Separable Space

Separable spaceFrom Wikipedia, the free encyclopedia

Contents

1 Axiom of countability 11.1 Important examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Relationships with each other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Baire space (set theory) 32.1 Topology and trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Relation to the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Banach space 53.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2.1 Linear operators, isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.2 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.3 Classical spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.4 Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.5 Dual space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.6 Banachs theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.7 Reexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.8 Weak convergences of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Schauder bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4.1 Tensor products and the approximation property . . . . . . . . . . . . . . . . . . . . . . . 153.5 Some classication results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5.1 Characterizations of Hilbert space among Banach spaces . . . . . . . . . . . . . . . . . . 153.5.2 Spaces of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.7 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

i

ii CONTENTS

3.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Bijection 214.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.1 Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.2 Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.6 Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.8 Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.9 Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Borel isomorphism 275.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Borel measure 286.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 On the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.3.1 LebesgueStieltjes integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3.2 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.3.3 Hausdor dimension and Frostmans lemma . . . . . . . . . . . . . . . . . . . . . . . . . 296.3.4 CramrWold theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 Cantor space 317.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

CONTENTS iii

7.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8 Cardinal number 338.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.3 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.4 Cardinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8.4.1 Successor cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.4.2 Cardinal addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.4.3 Cardinal multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.4.4 Cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8.5 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9 Cardinality of the continuum 419.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9.1.1 Uncountability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.1.2 Cardinal equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.1.3 Alternative explanation for c = 2@0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.2 Beth numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.3 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.4 Sets with cardinality of the continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.5 Sets with greater cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10 Closure (topology) 4610.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10.1.1 Point of closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.1.2 Limit point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.1.3 Closure of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.3 Closure operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.4 Facts about closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.5 Categorical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

11 Completely metrizable space 5011.1 Dierence between complete metric space and completely metrizable space . . . . . . . . . . . . . . 50

iv CONTENTS

11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.4 Completely metrizable abelian topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

12 Continuous function 5312.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5312.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

12.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5312.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.2.3 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.2.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.2.5 Directional and semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

12.3 Continuous functions between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.3.1 Uniform, Hlder and Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 63

12.4 Continuous functions between topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.4.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.4.3 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.4.4 Dening topologies via continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 67

12.5 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6812.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6812.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

13 Countable set 7013.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.4 Formal denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7113.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7613.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7613.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

14 Countably compact space 7814.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

CONTENTS v

14.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

15 Cover (topology) 7915.1 Cover in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.2 Renement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8015.4 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8015.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8015.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8115.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8115.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

16 Dense set 8216.1 Density in metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8216.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8216.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8316.4 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8316.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8316.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

16.6.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8416.6.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

17 Descriptive set theory 8517.1 Polish spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

17.1.1 Universality properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8517.2 Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

17.2.1 Borel hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8617.2.2 Regularity properties of Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

17.3 Analytic and coanalytic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8617.4 Projective sets and Wadge degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8617.5 Borel equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8717.6 Eective descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8717.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8717.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8717.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

18 Discrete space 8818.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9018.4 Indiscrete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9018.5 Quotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

vi CONTENTS

18.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9018.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

19 Ernst Leonard Lindelf 9119.1 Selected bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9119.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9119.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

20 Euclidean space 9320.1 Intuitive overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9320.2 Euclidean structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

20.2.1 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9520.2.2 Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.2.3 Rotations and reections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.2.4 Euclidean group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

20.3 Non-Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9820.4 Geometric shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

20.4.1 Lines, planes, and other subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9920.4.2 Line segments and triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10020.4.3 Polytopes and root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10120.4.4 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10120.4.5 Balls, spheres, and hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

20.5 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10220.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10220.7 Alternatives and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

20.7.1 Curved spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10220.7.2 Indenite quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10220.7.3 Other number elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10320.7.4 Innite dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

20.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10320.9 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10320.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10320.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

21 Exhaustion by compact sets 10421.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10421.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10421.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

22 Filter (mathematics) 10522.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10622.2 General denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10622.3 Filter on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

CONTENTS vii

22.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10722.3.2 Filters in model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10722.3.3 Filters in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

22.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11022.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11022.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

23 Homeomorphism 11123.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11123.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

23.2.1 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11223.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11323.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11323.5 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11423.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11423.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11423.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

24 Inner regular measure 11524.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11524.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11524.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11524.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

25 Interval (mathematics) 11725.1 Notations for intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

25.1.1 Including or excluding endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11725.1.2 Innite endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11825.1.3 Integer intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

25.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11825.3 Classication of intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

25.3.1 Intervals of the extended real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11925.4 Properties of intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11925.5 Dyadic intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12025.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

25.6.1 Multi-dimensional intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12025.6.2 Complex intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

25.7 Topological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12025.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12125.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12125.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

26 Johann Radon 122

viii CONTENTS

26.1 Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12226.2 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12226.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12326.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

27 Lawson topology 12427.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12427.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12427.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12427.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

28 Lexicographic order topology on the unit square 12528.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12528.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12528.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12528.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

29 Limit (mathematics) 12629.1 Limit of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12629.2 Limit of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12729.3 Limit as standard part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.4 Convergence and xed point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.5 Topological net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12929.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12929.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

30 Limit point 13030.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13030.2 Types of limit points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13030.3 Some facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13130.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13130.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

31 Lindelf space 13331.1 Properties of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13331.2 Properties of strongly Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13331.3 Product of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13331.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13431.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13431.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13431.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

32 Lindelfs lemma 135

CONTENTS ix

32.1 Statement of the lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13532.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13532.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

33 List of examples in general topology 13633.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

34 Local homeomorphism 13834.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13834.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13834.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13834.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13934.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

35 Local property 14035.1 Properties of a single space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

35.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14035.2 Properties of a pair of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14035.3 Properties of innite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14135.4 Properties of nite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14135.5 Properties of commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

36 Locally compact space 14236.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14236.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

36.2.1 Compact Hausdor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14336.2.2 Locally compact Hausdor spaces that are not compact . . . . . . . . . . . . . . . . . . . 14336.2.3 Hausdor spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 14336.2.4 Non-Hausdor examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

36.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14436.3.1 The point at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14436.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

36.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14536.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

37 Locally connected space 14637.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14737.2 Denitions and rst examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

37.2.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14837.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14837.4 Components and path components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

37.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14937.5 Quasicomponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

x CONTENTS

37.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14937.6 More on local connectedness versus weak local connectedness . . . . . . . . . . . . . . . . . . . . 15037.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15037.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15037.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15137.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

38 Locally nite collection 15238.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

38.1.1 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.1.2 Second countable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

38.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.3 Countably locally nite collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

39 Locally nite space 15439.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

40 Metric space 15540.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15540.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15540.3 Examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15640.4 Open and closed sets, topology and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 15740.5 Types of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

40.5.1 Complete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15740.5.2 Bounded and totally bounded spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15840.5.3 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15940.5.4 Locally compact and proper spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15940.5.5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15940.5.6 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

40.6 Types of maps between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15940.6.1 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16040.6.2 Uniformly continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16040.6.3 Lipschitz-continuous maps and contractions . . . . . . . . . . . . . . . . . . . . . . . . . 16040.6.4 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16140.6.5 Quasi-isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

40.7 Notions of metric space equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16140.8 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16140.9 Distance between points and sets; Hausdor distance and Gromov metric . . . . . . . . . . . . . . 16240.10Product metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

40.10.1 Continuity of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16240.11Quotient metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

CONTENTS xi

40.12Generalizations of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16340.12.1 Metric spaces as enriched categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

40.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16440.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16440.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16540.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

41 Metrization theorem 16641.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16641.2 Metrization theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16641.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16741.4 Examples of non-metrizable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16741.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16741.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

42 Neighbourhood (mathematics) 16842.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16942.2 In a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16942.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17142.4 Topology from neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17142.5 Uniform neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17142.6 Deleted neighbourhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17142.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17142.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

43 Neighbourhood system 17343.1 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17343.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17343.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17343.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17443.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

44 Net (mathematics) 17544.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17544.2 Examples of nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17544.3 Limits of nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17644.4 Examples of limits of nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17644.5 Supplementary denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17644.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17644.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17744.8 Cauchy nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17844.9 Relation to lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17844.10Limit superior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

xii CONTENTS

44.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

45 Paracompact space 18045.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18045.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18045.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18145.4 Paracompact Hausdor Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

45.4.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18245.5 Relationship with compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

45.5.1 Comparison of properties with compactness . . . . . . . . . . . . . . . . . . . . . . . . . 18345.6 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

45.6.1 Denition of relevant terms for the variations . . . . . . . . . . . . . . . . . . . . . . . . . 18445.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18445.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18445.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18545.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

46 Polish space 18646.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18646.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18746.3 Polish metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18746.4 Generalizations of Polish spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

46.4.1 Lusin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18746.4.2 Suslin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18746.4.3 Radon spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18846.4.4 Polish groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

46.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18846.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

47 Probability measure 18947.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19047.2 Example applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19047.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19147.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19147.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

48 Probability theory 19248.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19248.2 Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

48.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19248.2.2 Discrete probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19348.2.3 Continuous probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19448.2.4 Measure-theoretic probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

CONTENTS xiii

48.3 Classical probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19548.4 Convergence of random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

48.4.1 Law of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19648.4.2 Central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

48.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19748.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19748.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19848.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

49 Quotient space (topology) 19949.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20049.2 Quotient map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20049.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20049.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20149.5 Compatibility with other topological notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20249.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

49.6.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20249.6.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

49.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

50 Radon measure 20350.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20350.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20350.3 Radon measures on locally compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

50.3.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20450.3.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

50.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20550.5 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

50.5.1 Moderated Radon measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20650.5.2 Radon spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20650.5.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20650.5.4 Metric space structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

50.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20750.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

51 Radon space 20851.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

52 Second-countable space 20952.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

52.1.1 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20952.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21052.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

xiv CONTENTS

53 Separable space 21153.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21153.2 Separability versus second countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21153.3 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21253.4 Constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21253.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

53.5.1 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21253.5.2 Non-separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

53.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21353.6.1 Embedding separable metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

53.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

54 Subspace topology 21554.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21554.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21554.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21654.4 Preservation of topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21754.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21754.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

55 Topological space 21855.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

55.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21855.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21955.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22055.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

55.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22055.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22055.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22155.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22255.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22255.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22255.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22255.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22255.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22355.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22355.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22355.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

56 Uncountable set 22556.1 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22556.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

CONTENTS xv

56.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22556.4 Without the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22656.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22656.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22656.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

57 -compact space 22757.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22757.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22757.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22857.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22857.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 229

57.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22957.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23557.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

Chapter 1

Axiom of countability

In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) thatasserts the existence of a countable set with certain properties. Without such an axiom, such a set might not exist.

1.1 Important examplesImportant countability axioms for topological spaces include:[1]

sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set

rst-countable space: every point has a countable neighbourhood basis (local base)

second-countable space: the topology has a countable base

separable space: there exists a countable dense subspace

Lindelf space: every open cover has a countable subcover

-compact space: there exists a countable cover by compact spaces

1.2 Relationships with each otherThese axioms are related to each other in the following ways:

Every rst countable space is sequential.

Every second-countable space is rst-countable, separable, and Lindelf.

Every -compact space is Lindelf.

Every metric space is rst countable.

For metric spaces second-countability, separability, and the Lindelf property are all equivalent.

1.3 Related conceptsOther examples of mathematical objects obeying axioms of innity include sigma-nite measure spaces, and latticesof countable type.

1

2 CHAPTER 1. AXIOM OF COUNTABILITY

1.4 References[1] Nagata, J.-I. (1985), Modern General Topology, North-Holland Mathematical Library (3rd ed.), Elsevier, p. 104, ISBN

9780080933795.

Chapter 2

Baire space (set theory)

For the concept in topology, see Baire space.

In set theory, the Baire space is the set of all innite sequences of natural numbers with a certain topology. Thisspace is commonly used in descriptive set theory, to the extent that its elements are often called reals. It is oftendenoted B, NN, , or . Moschovakis denotes it N .The Baire space is dened to be the Cartesian product of countably innitely many copies of the set of naturalnumbers, and is given the product topology (where each copy of the set of natural numbers is given the discretetopology). The Baire space is often represented using the tree of nite sequences of natural numbers.The Baire space can be contrasted with Cantor space, the set of innite sequences of binary digits.

2.1 Topology and treesThe product topology used to dene the Baire space can be described more concretely in terms of trees. The denitionof the product topology leads to this characterization of basic open sets:

If any nite set of natural number coordinates {ci : i < n } is selected, and for each ci a particular naturalnumber value vi is selected, then the set of all innite sequences of natural numbers that have value viat position ci for all i < n is a basic open set. Every open set is a union of a collection of these.

By moving to a dierent basis for the same topology, an alternate characterization of open sets can be obtained:

If a sequence of natural numbers {wi : i < n} is selected, then the set of all innite sequences of naturalnumbers that have value wi at position i for all i < n is a basic open set. Every open set is a union of acollection of these.

Thus a basic open set in the Baire space species a nite initial segment of an innite sequence of natural numbers,and all the innite sequences extending form a basic open set. This leads to a representation of the Baire space asthe set of all paths through the full tree

4 CHAPTER 2. BAIRE SPACE (SET THEORY)

1. It is a perfect Polish space, which means it is a completely metrizable second countable space with no isolatedpoints. As such, it has the same cardinality as the real line and is a Baire space in the topological sense of theterm.

2. It is zero-dimensional and totally disconnected.3. It is not locally compact.

4. It is universal for Polish spaces in the sense that it can be mapped continuously onto any non-empty Polishspace. Moreover, any Polish space has a dense G subspace homeomorphic to a G subspace of the Bairespace.

5. The Baire space is homeomorphic to the product of any nite or countable number of copies of itself.

2.3 Relation to the real lineThe Baire space is homeomorphic to the set of irrational numbers when they are given the subspace topology inheritedfrom the real line. A homeomorphism between Baire space and the irrationals can be constructed using continuedfractions.From the point of view of descriptive set theory, the fact that the real line is connected causes technical diculties.For this reason, it is more common to study Baire space. Because every Polish space is the continuous image of Bairespace, it is often possible to prove results about arbitrary Polish spaces by showing that these properties hold for Bairespace and by showing that they are preserved by continuous functions.B is also of independent, but minor, interest in real analysis, where it is considered as a uniform space. The uniformstructures ofB and Ir (the irrationals) are dierent, however: B is complete in its usual metric while Ir is not (althoughthese spaces are homeomorphic).

2.4 References Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 0-387-94374-9. Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

Chapter 3

Banach space

In mathematics, more specically in functional analysis, aBanach space (pronounced [banax]) is a complete normedvector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length anddistance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a welldened limit that is within the space.Banach spaces are named after the Polish mathematician Stefan Banach, who introduced and made a systematic studyof them in 19201922 along with Hans Hahn and Eduard Helly.[1] Banach spaces originally grew out of the study offunction spaces by Hilbert, Frchet, and Riesz earlier in the century. Banach spaces play a central role in functionalanalysis. In other areas of analysis, the spaces under study are often Banach spaces.

3.1 DenitionA Banach space is a vector space X over the eld R of real numbers, or over the eld C of complex numbers, whichis equipped with a norm and which is complete with respect to that norm, that is to say, for every Cauchy sequence{xn} in X, there exists an element x in X such that

limn!1xn = x;

or equivalently:

limn!1 kxn xkX = 0:

The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series ofvectors. A normed space X is a Banach space if and only if each absolutely convergent series in X converges,[2]

1Xn=1

kvnkX

6 CHAPTER 3. BANACH SPACE

3.2.1 Linear operators, isomorphismsMain article: Bounded operator

If X and Y are normed spaces over the same ground eld K, the set of all continuous K-linear maps T : X Yis denoted by B(X, Y). In innite-dimensional spaces, not all linear maps are continuous. A linear mapping from anormed space X to another normed space is continuous if and only if it is bounded on the closed unit ball of X. Thus,the vector space B(X, Y) can be given the operator norm

kTk = sup fkTxkY j x 2 X; kxkX 1g :For Y a Banach space, the space B(X, Y) is a Banach space with respect to this norm.If X is a Banach space, the space B(X) = B(X, X) forms a unital Banach algebra; the multiplication operation is givenby the composition of linear maps.If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T : X Y suchthat T and its inverse T 1 are continuous. If one of the two spaces X or Y is complete (or reexive, separable, etc.)then so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry,i.e., ||T(x)|| = ||x|| for every x in X. The Banach-Mazur distance d(X, Y) between two isomorphic but not isometricspaces X and Y gives a measure of how much the two spaces X and Y dier.

3.2.2 Basic notionsEvery normed space X can be isometrically embedded in a Banach space. More precisely, there is a Banach space Yand an isometric mapping T : X Y such that T(X) is dense in Y. If Z is another Banach space such that there is anisometric isomorphism from X onto a dense subset of Z, then Z is isometrically isomorphic to Y.This Banach space Y is the completion of the normed space X. The underlying metric space for Y is the same asthe metric completion of X, with the vector space operations extended from X to Y. The completion of X is oftendenoted by bX .The cartesian product X Y of two normed spaces is not canonically equipped with a norm. However, severalequivalent norms are commonly used,[4] such as

k(x; y)k1 = kxk+ kyk; k(x; y)k1 = max(kxk; kyk)and give rise to isomorphic normed spaces. In this sense, the product X Y (or the direct sum X Y) is complete ifand only if the two factors are complete.If M is a closed linear subspace of a normed space X, there is a natural norm on the quotient space X / M,

kx+Mk = infm2M

kx+mk:

The quotient X / M is a Banach space when X is complete.[5] The quotient map from X onto X / M, sending x in Xto its class x + M, is linear, onto and has norm 1, except when M = X, in which case the quotient is the null space.The closed linear subspace M of X is said to be a complemented subspace of X if M is the range of a boundedlinear projection P from X onto M. In this case, the space X is isomorphic to the direct sum of M and Ker(P), thekernel of the projection P.Suppose that X and Y are Banach spaces and that T B(X, Y). There exists a canonical factorization of T as[5]

T = T1 ; T : X ! X/Ker(T ) T1! Ywhere the rst map is the quotient map, and the second map T1 sends every class x + Ker(T) in the quotient to theimage T(x) in Y. This is well dened because all elements in the same class have the same image. The mapping T1is a linear bijection from X / Ker(T) onto the range T(X), whose inverse need not be bounded.

3.2. GENERAL THEORY 7

3.2.3 Classical spacesBasic examples[6] of Banach spaces include: the Lp spaces and their special cases, the sequence spaces p that consistof scalar sequences indexed by N; among them, the space 1 of absolutely summable sequences and the space 2 ofsquare summable sequences; the space c0 of sequences tending to zero and the space of bounded sequences; thespace C(K) of continuous scalar functions on a compact Hausdor space K, equipped with the max norm,

kfkC(K) = maxfjf(x)j : x 2 Kg; f 2 C(K):According to the BanachMazur theorem, every Banach space is isometrically isomorphic to a subspace of someC(K).[7] For every separable Banach space X, there is a closed subspace M of 1 such that X 1/M.[8]

Any Hilbert space serves as an example of a Banach space. A Hilbert space H on K = R, C is complete for a normof the form

kxkH =phx; xi;

where

h; i : H H ! Kis the inner product, linear in its rst argument that satises the following:

8x; y 2 H : hy; xi = hx; yi;8x 2 H : hx; xi 0;hx; xi = 0, x = 0:

For example, the space L2 is a Hilbert space.The Hardy spaces, the Sobolev spaces are examples of Banach spaces that are related to Lp spaces and have additionalstructure. They are important in dierent branches of analysis, Harmonic analysis and Partial dierential equationsamong others.

3.2.4 Banach algebrasA Banach algebra is a Banach space A over K = R or C, together with a structure of algebra over K, such that theproduct map (a, b) A A ab A is continuous. An equivalent norm on A can be found so that ||ab|| ||a|| ||b||for all a, b A.

Examples

The Banach space C(K), with the pointwise product, is a Banach algebra. The disk algebra A(D) consists of functions holomorphic in the open unit disk D C and continuous on its

closure: D. Equipped with the max norm on D, the disk algebra A(D) is a closed subalgebra of C(D). The Wiener algebra A(T) is the algebra of functions on the unit circle T with absolutely convergent Fourier

series. Via the map associating a function on T to the sequence of its Fourier coecients, this algebra isisomorphic to the Banach algebra 1(Z), where the product is the convolution of sequences.

For every Banach space X, the space B(X) of bounded linear operators on X, with the composition of maps asproduct, is a Banach algebra.

A C*-algebra is a complex Banach algebra A with an antilinear involution a a such that ||aa|| = ||a||2.The space B(H) of bounded linear operators on a Hilbert space H is a fundamental example of C*-algebra.The GelfandNaimark theorem states that every C*-algebra is isometrically isomorphic to a C*-subalgebra ofsome B(H). The space C(K) of complex continuous functions on a compact Hausdor space K is an exampleof commutative C*-algebra, where the involution associates to every function f its complex conjugate f .


3.2.5 Dual spaceMain article: Dual space

If X is a normed space and K the underlying eld (either the real or the complex numbers), the continuous dualspace is the space of continuous linear maps from X into K, or continuous linear functionals. The notation for thecontinuous dual is X = B(X, K) in this article.[9] Since K is a Banach space (using the absolute value as norm), thedual X is a Banach space, for every normed space X.The main tool for proving the existence of continuous linear functionals is the HahnBanach theorem.

HahnBanach theorem. Let X be a vector space over the eld K = R, C. Let further

Y X be a linear subspace, p : X R be a sublinear function and f : Y K be a linear functional so that Re( f (y)) p(y) for all y in Y.

Then, there exists a linear functional F : X K so that

F jY = f; and 8x 2 X; Re(F (x)) p(x):

In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to thewhole space, without increasing the norm of the functional.[10] An important special case is the following: for everyvector x in a normed space X, there exists a continuous linear functional f on X such that

f(x) = kxkX ; kfkX0 1:

When x is not equal to the 0 vector, the functional f must have norm one, and is called a norming functional for x.The HahnBanach separation theorem states that two disjoint non-empty convex sets in a real Banach space, oneof them open, can be separated by a closed ane hyperplane. The open convex set lies strictly on one side of thehyperplane, the second convex set lies on the other side but may touch the hyperplane.[11]

A subset S in a Banach space X is total if the linear span of S is dense in X. The subset S is total in X if andonly if the only continuous linear functional that vanishes on S is the 0 functional: this equivalence follows from theHahnBanach theorem.If X is the direct sum of two closed linear subspaces M and N, then the dual X of X is isomorphic to the direct sumof the duals of M and N.[12] If M is a closed linear subspace in X, one can associate the orthogonal of M in the dual,

M? = fx0 2 X 0 : x0(m) = 0; 8m 2Mg :

The orthogonal M is a closed linear subspace of the dual. The dual of M is isometrically isomorphic to X / M .The dual of X / M is isometrically isomorphic to M .[13]

The dual of a separable Banach space need not be separable, but:

Theorem.[14] Let X be a normed space. If X is separable, then X is separable.

When X is separable, the above criterion for totality can be used for proving the existence of a countable total subsetin X.

Weak topologies

The weak topology on a Banach space X is the coarsest topology on X for which all elements x in the continuousdual space X are continuous. The norm topology is therefore ner than the weak topology. It follows from theHahnBanach separation theorem that the weak topology is Hausdor, and that a norm-closed convex subset of a


Banach space is also weakly closed.[15] A norm-continuous linear map between two Banach spaces X and Y is alsoweakly continuous, i.e., continuous from the weak topology of X to that of Y.[16]

If X is innite-dimensional, there exist linear maps which are not continuous. The space X of all linear maps fromX to the underlying eld K (this space X is called the algebraic dual space, to distinguish it from X ) also induces atopology on X which is ner than the weak topology, and much less used in functional analysis.On a dual space X , there is a topology weaker than the weak topology of X , called weak* topology. It is thecoarsest topology on X for which all evaluation maps x X x(x), x X, are continuous. Its importance comesfrom the BanachAlaoglu theorem.

BanachAlaoglu Theorem. Let X be a normed vector space. Then the closed unit ball B = {x X : ||x|| 1} of the dual space is compact in the weak* topology.

The BanachAlaoglu theorem depends on Tychonos theorem about innite products of compact spaces. When Xis separable, the unit ball B of the dual is a metrizable compact in the weak* topology.[17]

Examples of dual spaces

The dual of c0 is isometrically isomorphic to 1: for every bounded linear functional f on c0, there is a unique elementy = {yn} 1 such that

f(x) =Xn2N

xnyn; x = fxng 2 c0; and kfk(c0)0 = kyk`1 :

The dual of 1 is isometrically isomorphic to . The dual of Lp([0, 1]) is isometrically isomorphic to Lq([0, 1])when 1 p < and 1/p + 1/q = 1.For every vector y in a Hilbert space H, the mapping

x 2 H ! fy(x) = hx; yidenes a continuous linear functional fy on H. The Riesz representation theorem states that every continuous linearfunctional on H is of the form fy for a uniquely dened vector y in H. The mapping y H fy is an antilinearisometric bijection from H onto its dual H . When the scalars are real, this map is an isometric isomorphism.When K is a compact Hausdor topological space, the dualM(K) of C(K) is the space of Radon measures in the senseof Bourbaki.[18] The subset P(K) of M(K) consisting of non-negative measures of mass 1 (probability measures) is aconvex w*-closed subset of the unit ball of M(K). The extreme points of P(K) are the Dirac measures on K. The setof Dirac measures on K, equipped with the w*-topology, is homeomorphic to K.

Banach-Stone Theorem. If K and L are compact Hausdor spaces and if C(K) and C(L) are isomet-rically isomorphic, then the topological spaces K and L are homeomorphic.[19][20]

The result has been extended by Amir[21] and Cambern[22] to the case when the multiplicative BanachMazur distancebetween C(K) and C(L) is < 2. The theorem is no longer true when the distance is = 2.[23]

In the commutative Banach algebra C(K), the maximal ideals are precisely kernels of Dirac mesures on K,

Ix = ker x = ff 2 C(K) : f(x) = 0g; x 2 K:More generally, by the Gelfand-Mazur theorem, the maximal ideals of a unital commutative Banach algebra can beidentied with its characters---not merely as sets but as topological spaces: the former with the hull-kernel topologyand the latter with the w*-topology. In this identication, the maximal ideal space can be viewed as a w*-compactsubset of the unit ball in the dual A .

Theorem. If K is a compact Hausdor space, then the maximal ideal space of the Banach algebraC(K) is homeomorphic to K.[19]


Not every unital commutative Banach algebra is of the form C(K) for some compact Hausdor space K. However,this statement holds if one places C(K) in the smaller category of commutative C*-algebras. Gelfands representationtheorem for commutative C*-algebras states that every commutative unital C*-algebra A is isometrically isomorphicto a C(K) space.[24] The Hausdor compact space K here is again the maximal ideal space, also called the spectrumof A in the C*-algebra context.

Bidual

If X is a normed space, the (continuous) dual X of the dual X is called bidual, or second dual of X. For everynormed space X, there is a natural map,

(FX : X ! X 00FX(x)(f) = f(x) 8x 2 X;8f 2 X 0

This denes FX(x) as a continuous linear functional on X , i.e., an element of X . The map FX : x FX(x) is alinear map from X to X . As a consequence of the existence of a norming functional f for every x in X, this mapFX is isometric, thus injective.For example, the dual of X = c0 is identied with 1, and the dual of 1 is identied with , the space of boundedscalar sequences. Under these identications, FX is the inclusion map from c0 to . It is indeed isometric, but notonto.If FX is surjective, then the normed space X is called reexive (see below). Being the dual of a normed space, thebidual X is complete, therefore, every reexive normed space is a Banach space.Using the isometric embedding FX, it is customary to consider a normed space X as a subset of its bidual. When Xis a Banach space, it is viewed as a closed linear subspace of X . If X is not reexive, the unit ball of X is a propersubset of the unit ball of X . The Goldstine theorem states that the unit ball of a normed space is weakly*-dense inthe unit ball of the bidual. In other words, for every x in the bidual, there exists a net {xj} in X so that

supjkxjk kx00k; x00(f) = lim

jf(xj); f 2 X 0:

The net may be replaced by a weakly*-convergent sequence when the dual X is separable. On the other hand,elements of the bidual of 1 that are not in 1 cannot be weak*-limit of sequences in 1, since 1 is weakly sequentiallycomplete.

3.2.6 Banachs theorems

Here are the main general results about Banach spaces that go back to the time of Banachs book (Banach (1932))and are related to the Baire category theorem. According to this theorem, a complete metric space (such as a Banachspace, a Frchet space or an F-space) cannot be equal to a union of countably many closed subsets with emptyinteriors. Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is alreadyequal to one of them; a Banach space with a countable Hamel basis is nite-dimensional.

BanachSteinhaus Theorem. Let X be a Banach space and Y be a normed vector space. Suppose thatF is a collection of continuous linear operators from X to Y. The uniform boundedness principle statesthat if for all x in X we have supTF ||T(x)||Y < , then supTF ||T ||Y < .

The BanachSteinhaus theorem is not limited to Banach spaces. It can be extended for example to the case whereX is a Frchet space, provided the conclusion is modied as follows: under the same hypothesis, there exists aneighborhood U of 0 in X such that all T in F are uniformly bounded on U,

supT2F

supx2U

kT (x)kY


The Open Mapping Theorem. Let X and Y be Banach spaces and T : X Y be a continuous linearoperator. Then T is surjective if and only if T is an open map.

Corollary. Every one-to-one bounded linear operator from a Banach space onto a Banach space is anisomorphism.

The First Isomorphism Theorem for Banach spaces. Suppose that X and Y are Banach spaces andthat T B(X, Y). Suppose further that the range of T is closed in Y. Then X/ Ker(T) is isomorphic toT(X).

This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorizationof bounded linear maps.

Corollary. If a Banach space X is the internal direct sum of closed subspaces M1, ..., Mn, then X isisomorphic to M1 ... Mn.

This is another consequence of Banachs isomorphism theorem, applied to the continuous bijection from M1 ... Mn onto X sending (m1, ..., mn) to the sum m1 + ... + mn.

The Closed Graph Theorem. Let T : X Y be a linear mapping between Banach spaces. The graphof T is closed in X Y if and only if T is continuous.

3.2.7 ReexivityMain article: Reexive space

The normed space X is called reexive when the natural map

(FX : X ! X 00FX(x)(f) = f(x) 8x 2 X;8f 2 X 0

is surjective. Reexive normed spaces are Banach spaces.

Theorem. If X is a reexive Banach space, every closed subspace of X and every quotient space of Xare reexive.

This is a consequence of the HahnBanach theorem. Further, by the open mapping theorem, if there is a boundedlinear operator from the Banach space X onto the Banach space Y, then Y is reexive.

Theorem. If X is a Banach space, then X is reexive if and only if X is reexive.

Corollary. Let X be a reexive Banach space. Then X is separable if and only if X is separable.

Indeed, if the dual Y of a Banach space Y is separable, then Y is separable. If X is reexive and separable, then thedual of X is separable, so X is separable.

Theorem. Suppose that X1, ..., Xn are normed spaces and that X = X1 ... Xn. Then X is reexiveif and only if each Xj is reexive.

Hilbert spaces are reexive. The Lp spaces are reexive when 1 < p < . More generally, uniformly convex spaces arereexive, by the MilmanPettis theorem. The spaces c0, 1, L1([0, 1]), C([0, 1]) are not reexive. In these examplesof non-reexive spaces X, the bidual X is much larger than X. Namely, under the natural isometric embedding ofX into X given by the HahnBanach theorem, the quotient X / X is innite-dimensional, and even nonseparable.However, Robert C. James has constructed an example[25] of a non-reexive space, usually called "the James space"and denoted by J,[26] such that the quotient J / J is one-dimensional. Furthermore, this space J is isometricallyisomorphic to its bidual.


Theorem. A Banach space X is reexive if and only if its unit ball is compact in the weak topology.

When X is reexive, it follows that all closed and bounded convex subsets of X are weakly compact. In a Hilbertspace H, the weak compactness of the unit ball is very often used in the following way: every bounded sequence inH has weakly convergent subsequences.Weak compactness of the unit ball provides a tool for nding solutions in reexive spaces to certain optimizationproblems. For example, every convex continuous function on the unit ball B of a reexive space attains its minimumat some point in B.As a special case of the preceding result, when X is a reexive space over R, every continuous linear functional f inX attains its maximum || f || on the unit ball of X. The following theorem of Robert C. James provides a conversestatement.

James Theorem. For a Banach space the following two properties are equivalent: X is reexive. for all f in X there exists x in X with ||x|| 1, so that f (x) = || f ||.

The theorem can be extended to give a characterization of weakly compact convex sets.On every non-reexive Banach space X, there exist continuous linear functionals that are not norm-attaining. How-ever, the BishopPhelps theorem[27] states that norm-attaining functionals are norm dense in the dual X of X.

3.2.8 Weak convergences of sequencesA sequence {xn} in a Banach space X is weakly convergent to a vector x X if f (xn) converges to f (x) for everycontinuous linear functional f in the dual X . The sequence {xn} is a weakly Cauchy sequence if f (xn) converges toa scalar limit L( f ), for every f in X . A sequence { fn } in the dual X is weakly* convergent to a functional f X if fn (x) converges to f (x) for every x in X. Weakly Cauchy sequences, weakly convergent and weakly* convergentsequences are norm bounded, as a consequence of the BanachSteinhaus theorem.When the sequence {xn} in X is a weakly Cauchy sequence, the limit L above denes a bounded linear functionalon the dual X , i.e., an element L of the bidual of X, and L is the limit of {xn} in the weak*-topology of the bidual.The Banach space X is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent in X.It follows from the preceding discussion that reexive spaces are weakly sequentially complete.

Theorem. [28] For every measure , the space L1() is weakly sequentially complete.

An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equalto the 0 vector. The unit vector basis of p, 1 < p < , or of c0, is another example of a weakly null sequence, i.e.,a sequence that converges weakly to 0. For every weakly null sequence in a Banach space, there exists a sequence ofconvex combinations of vectors from the given sequence that is norm-converging to 0.[29]

The unit vector basis of 1 is not weakly Cauchy. Weakly Cauchy sequences in 1 are weakly convergent, sinceL1-spaces are weakly sequentially complete. Actually, weakly convergent sequences in 1 are norm convergent.[30]This means that 1 satises Schurs property.

Results involving the 1 basis

Weakly Cauchy sequences and the 1 basis are the opposite cases of the dichotomy established in the following deepresult of H. P. Rosenthal.[31]

Theorem.[32] Let {xn} be a bounded sequence in a Banach space. Either {xn} has a weakly Cauchysubsequence, or it admits a subsequence equivalent to the standard unit vector basis of 1.

A complement to this result is due to Odell and Rosenthal (1975).

Theorem.[33] Let X be a separable Banach space. The following are equivalent:

3.3. SCHAUDER BASES 13

The space X does not contain a closed subspace isomorphic to 1. Every element of the bidual X is the weak*-limit of a sequence {xn} in X.

By the Goldstine theorem, every element of the unit ball B of X is weak*-limit of a net in the unit ball of X.When X does not contain 1, every element of B is weak*-limit of a sequence in the unit ball of X.[34]

When the Banach space X is separable, the unit ball of the dualX , equipped with the weak*-topology, is a metrizablecompact space K,[17] and every element x in the bidual X denes a bounded function on K:

x0 2 K 7! x00(x0); jx00(x0)j kx00k :

This function is continuous for the compact topology of K if and only if x is actually in X, considered as subset ofX . Assume in addition for the rest of the paragraph that X does not contain 1. By the preceding result of Odelland Rosenthal, the function x is the pointwise limit on K of a sequence {xn} X of continuous functions on K, it istherefore a rst Baire class function on K. The unit ball of the bidual is a pointwise compact subset of the rst Baireclass on K.[35]

Sequences, weak and weak* compactness

When X is separable, the unit ball of the dual is weak*-compact by BanachAlaoglu and metrizable for the weak*topology,[17] hence every bounded sequence in the dual has weakly* convergent subsequences. This applies to sepa-rable reexive spaces, but more is true in this case, as stated below.The weak topology of a Banach space X is metrizable if and only if X is nite-dimensional.[36] If the dual X isseparable, the weak topology of the unit ball of X is metrizable. This applies in particular to separable reexiveBanach spaces. Although the weak topology of the unit ball is not metrizable in general, one can characterize weakcompactness using sequences.

Eberleinmulian theorem.[37] A set A in a Banach space is relatively weakly compact if and only ifevery sequence {an} in A has a weakly convergent subsequence.

A Banach space X is reexive if and only if each bounded sequence in X has a weakly convergent subsequence.[38]

A weakly compact subset A in 1 is norm-compact. Indeed, every sequence in A has weakly convergent subsequencesby Eberleinmulian, that are norm convergent by the Schur property of 1.

3.3 Schauder basesMain article: Schauder basis

A Schauder basis in a Banach space X is a sequence {en}n of vectors in X with the property that for every vectorx in X, there exist uniquely dened scalars {xn}n depending on x, such that

x =1Xn=0

xnen; i.e., x = limnPn(x); Pn(x) :=

nXk=0

xkek:

Banach spaces with a Schauder basis are necessarily separable, because the countable set of nite linear combinationswith rational coecients (say) is dense.It follows from the BanachSteinhaus theorem that the linear mappings {Pn} are uniformly bounded by some constantC. Let {en} denote the coordinate functionals which assign to every x in X the coordinate xn of x in the above expansion. Theyare called biorthogonal functionals. When the basis vectors have norm 1, the coordinate functionals {en} have norm 2C in the dual of X.


Most classical separable spaces have explicit bases. The Haar system {hn} is a basis for Lp([0, 1]), 1 p

Documents

Separable Space