TALLERES ESTUDIANTILES CIENCIAS UNAM
REAL ANALYSISSECOND EDITION
Edicin digital:Educacin
para todos
Edicin impresa:
Macmillan Second edition Stanford U.
Educacin
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Contents
Prologue to the Student
1 Set Theory1 Introduction, 5 2 Functions, 8 3 Unions,
intersections, and complements, 11 4 Algebras of sets, 16 5 The
axiom of choice and infinite direct products, 18 6 Countable sets,
19 7 Relations and equivalences, 22 8 Partial orderings and the
maximal principle, 23 9 Well ordering and the countable ordinals,
24
Part One
Theory of Functions of a Real Variable29
2 The Real Number System1 2 3 4 5 6 7
Axioms for the real numbers, 29 The natural and rational numbers
as subsets of R, 32 The extended real numbers, 34 Sequences of real
numbers, 35 Open and closed sets o j real numbers, 38
Continuousjunctions, 44 Bore1 sets, 50 ix
Contents
x i142
8 Topological Spaces1 Fundamental notions, 142 2 Bases and
countability, 145 3 The separation axioms and continuous
real-valuedfunctions, 147 4 Product spaces, 150 5 Connectedness,
150 "6 Absolute SS'S,154 "7 Nets, 155
9 Compact Spaces1 2 3 4 5 "6 7 "8
157
Basic properties, 157 Countable compactness and the Bolzano-
Weierstrass property, 159 Compact metric spaces, 163 Products of
compact spaces, 165 Locally compact spaces, 168 The Stone-eech
compactijication, 170 The Stone- Weierstrass theorem, 171 The
Ascoli theorem, 177
10 Banach Spaces1 Introduction, 181 2 Linear operators, 184 3
Linear.functionals and the Hahn-Banach theorem, 186 4 The closed
graph theorem, 193 "5 Topological vector spaces, 197 "6 Weak
topologies, 200 "7 Convexity, 203 8 Hilbert space, 210
181
Part Three
0
General Measure and Integration Theory217
1 Measure and Integration 11 2 3 "4 5 6 7
Measure spaces, 217 MeasurabIe.functions, 223 Integration, 225
General convergence theorems, 231 Signed measures, 232 The
Radon-Nikodym theorem, 238 The L p spaces, 243
xii
Contents
12 Measure and Outer Measure1 Outer measure and measurability,
250 2 The extension theorem, 253 * 3 The Lebesgue-Stieltjes
integral, 26 1 4 Product measures, 264 " 5 Inner measure, 274 "6
Extension by sets qf measure zero, 281 "7 Carathbodory outer
measure, 283
250
13 The Daniel1 Integral1 2 3 4
286
Introduction, 286 The extension theorem, 288 Uniqueness, 294
Measurability and measure, 295
14 Measure and Topology1 2 3 "4
30 1
Baire sets and Borel sets, 301 Positive linear functionals and
Baire measures, 304 Bounded 1inear.functionalson C ( X ) , 308 The
Borel extension qf a measure, 313
15 Mappings of Measure Spaces1 Point mappings and set mappings,
317 2 Measure algebras, 319 3 Borel equivalences, 324 4 Set
mappings andpoint mappings on complete metric spaces, 328 5 The
isometries oj' L p , 331
317
Epilogue Bibliography Index of Symbols Subject Index
335 337 339 34 1
