List of Notation

The three numbers following a symbol denote the numbers of the chapter, section, and subsection, respectively, in the Theory part in which the meaning of this symbol is explained.

C is the set of complex numbers; N is the set of natural numbers; Q is the set of rational numbers; R is the set of real numbers; Z is the set of integers.

Spaces

A 2(D), 111.4.1, the space of analytic functions in the unit disk D; B(X), 111.3.2, the space of bounded functions on the set X; C(X), 111.1.4, the space of continuous functions on the set X; C(O)(C(Q)), 111.3.3, the space of r times continuously differentiable

functions in 0 (respectively, that extend continuously to Q); .@(0) = C0(0), 111.3.3, the space of compactly supported infinitely

differentiable functions on 0; .@'(0), 111.3.4, the space of generalized functions; $(0) = C 00(0), 111.3.3, the space of infinitely differentiable functions on

o· ' C'(O), 111.3.4, the space of generalized functions with compact support; End L = st'(L, L); !F(L1 , L 2 ), 111.2.3, the space of Fredholm operators from the LTS L 1 to

the LTS L 2 ;

335

336 List of Notation

H*, III.4.2, the Hermitian dual space of the Hilbert space H; H 1 EB H 2 , 111.4.1, direct sum of linear spaces; H 1 ® H 2 , 111.1.4, tensor product of linear spaces; .ff(L1, L2 ), 111.2.2, the space of compact operators from the LTS L1 to

the LTS L2 ;

!t?(L1, L2 ), 111.1.2, the space of continuous linear mappings from the LTS L1 totheLTSL2 ;

L', 111.1.2, the dual space of the L TS L; Lp(X, Jl), 111.3.1, the space of functions on X with 11-integrable pth powers; L 00 (X, Jl), 111.3.1, the space of essentially bounded functions on X; lp(n, K), 111.1.4, then-dimensional space over the field K with norm llxiiP; lp(K), 111.1.4, the space of sequences over the field K with norm llxiiP; p(Z), III.2.2, the space of slowly increasing sequences; P6"(Rn), IV1.2, the subspace of 6'(Rn) consisting of functions of not

greater than polynomial growth; S(Rn), 111.3.3, the space of smooth rapidly decreasing functions; S'(Rn), III.3.4, the space of tempered distributions; BV[a, b], 11.1.3, the space of functions of bounded variation on [a, b].

Convergence

An=> A, A = u-lim An, 111.2.1, uniform convergence of operators; An--+ A, A = s-lim An, 111.2.1, strong convergence of operators; An___. A, A = w-lim An, 111.2.1, weak convergence of operators; fn => f, 11.2.2, uniform convergence of functions; fn '::;; f, 11.2.2, convergence of functions almost everywhere; fn 4 f, 11.2.2, convergence of functions in the measure Jl·

Operators

A', 111.1.2, the operator ad joint to the operator A; A*, 111.4.2, the Hermitian adjoint operator of the operator A; A ~ 0, 111.4.2, a positive operator A; A, V.l.3, the closure of the operator A; A1 EB A 2 , 111.2.3, the direct sum of the operators A1 and A 2 ;

B ::::J A, V.l.3, the operator B is an extension of the operator A; coker A, 111.2.3, the co kernel of the operator A; D A, 111.2.3, the domain of the operator A; a, 111.3.3, a partial derivative operator; i(A), 111.2.3, the index of the operator A; im A, 111.2.3, the range of the operator A; ker A, 111.2.3, the kernel of the operator A; M(f), 111.3.5, the operator of multiplication by the function f; r(A), V.l.2, the spectral radius of the operator A;

List of Notation

r;.(A), V.l.l, the resolvent of the operator A; rank A, III.2.3, the rank of the operator A; S(f), IV.l.l, the operator of convolution with the function f; T(a), IV.l.l, the operator of translation by a; p(A), V.l.2, the resolvent set of the operator A; u(A), V.l.2, the spectrum of the operator A.

Other Notation

337

A 1::,. B, 11.1.1, the symmetric difference of the sets A and B; AU B (or UA;), Il.l.l, disjoint union of sets; B(x, r) (B(x, r)), III.l.l, the open (resp., closed) ball of radius r with

center at the point x in a metric space; Card Y, 11.1.3, the number of elements of the set Y; Cont(K 1, K2 ) (Cov(K 1, K 2 )), 1.3, the contravariant (resp., covariant)

functors; diam X, III.2.2, the diameter of the set X; ess sup J, 11.3.1, the essential supremum of the function f; ], IV.2.1, the Fourier transform of the function f; f 1 x f 2 , 111.3.5, the direct product of the generalized functions f 1 and f 2 ;

f 1 * f 2 , IV.l.l, the convolution of the functions f 1 and f 2 ;

G, IV.2.1, the dual group of the group G; K[G], IV.l.l, the group algebra of the group G; K 0 , 1.3, the category dual to the category K; L(S, Jl), 11.1.2, the collection of }1-measurable sets; Mor(A, B), Ob(K), 1.3, the morphisms and objects in the category K; P(X), 11.1.1, the set of subsets of X; QA(X), III.4.2, the Hermitian (quadratic) form corresponding to the

operator A; R(S), 11.1.1, the ring generated by the family of sets S; Ru(S), 11.1.1, the u-ring generated by the family of sets S; supp <p, 111.3.3, the support of the function <p; T", IV.l.l, then-dimensional torus; Var~ f, Il.l.3, the variation of the function f on the segment [a, b]; X\ III.2.3, the orthogonal complement of X; x l_ y, III.4.1, the vector x is orthogonal to the vector y; b(x), Ill.3.4, the Dirac function (<>-function); bb(x), 111.3.5, the translated <>-function; bij, 111.2.3, the Kronecker symbols; Jl( A), 11.1.2, the measure of the set A ; Jl*(A), 11.1.2, the outer measure of the set A; v(A), 11.1.2, a signed measure; I vI (A), 11.1.2, the variation of the signed measure v; <P(x), IV.l.2, <P(x) = <p( -x); XA, 11.1.1, the characteristic function of the set A.

Bibliography

Basic Literature

I. N. I. Akhiezer and I. M. Glazman. The Theory of Linear Operators in Hilbert Space, Vols. I, II. New York: Ungar, 1961.

2. F. A. Berezin, A. D. Gvishiani, E. A. Gorin, and A. A. Kirillov, Collection of Problems in Functional Analysis. Moscow: Moscow State Univ .. 1977 (Russian).

3. N. Bourbaki. Integration, Chapitres 1-5. Actualites Sci. Indust., nos. 1175, 1244, Paris: Hermann, 1952, 1956.

4. N. Bourbaki. Integration, Chapitres 3-5, 9. Actualites Sci. Indust., nos. 1175 (2nd ed.), 1244 (2nd ed.), 1343. Paris: Hermann, 1965, 1967, 1969.

5. N. Bourbaki. Theorie Spectrales, Chapitres I, 2, Actualites Sci. Indust., no. 1332, Paris: Hermann, 1967.

6. N. Bourbaki. Theory of Sets. Reading, Mass.: Addison-Wesley, 1968. 7. N. Bourbaki. Espaces Vectoriels Topologiques, Chapitres 1-5 et fascicule de

resultats. Actualites Sci. Indust., nos. 1189, 1229, 1230. Paris: Hermann, 1953, 1955.

8. V. S. Vladimirov. Generalized Functions in Mathematical Physics. Moscow: Mir, 1979.

9. V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, Kh. Kh. Karimova, Yu. V. Sidorov, and M. I. Shabunin. Collection of Problems on the Equations of Mathe­matical Physics. Moscow: Mir, 1976 (French translation).

10. I. M. Gel'fand and G. E. Shilov. Generalized Functions, Vol. I: Generalized Func­tions and Operations on Them. New York: Academic Press, 1964.

II. I. M. Gel'fand and G. E. Shilov. Generalized Functions, Vol. 2: Spaces of Test Functions and Generalized Functions. New York: Academic Press; Gordon & Breach, 1958.

12. I. M. Gel'fand and N. Ya. Vilenkin. Generalized Functions, Vol. 4: Some Applica­tions of Harmonic Analysis. New York: Academic Press, 1964.

13. N. Dunford and J. T. Schwartz. Linear Operators, Vol. I. New York: Interscience, 1958.

339

340 Bibliography

14. N. Dunford and J. T. Schwartz. Linear Operators, Vol. II. New York: Inter­science, 1963.

15. K. Yoshida. Functional Analysis. Berlin: Springer-Verlag, 1965. 16. L. V. Kantorovich and G. P. Akilov. Functional Analysis, 2nd ed. Moscow: Nauka,

1977 (Russian). 17. A. A. Kirillov, Elements of the Theory of Representations. Berlin-New York:

Springer-Verlag, 1976. 18. A. N. Kolmogorov and S. V. Fomin. Elements of the Theory of Functions and

Functional Analysis, Vo1s. I, II. Albany, N.Y.: Graylock Press, 1957, 1961. 19. L. D. Kudryavtsev. Mathematical Analysis, Vols. I, 2, Moscow: Vysshaya Shkola,

1963, 1970 (Russian). 20. K. Maurin. Metody Przestrzeni Hilbert (Hilbert Space Methods). Polska Akademia

Nauk, Monografie Mat., Vol. 36. Warsaw: PWN, 1959 (Polish). 21. M. A. Naimark. Linear Differential Operators, Parts I, II. New York: Ungar,

1967, 1968. 22. L. S. Pontryagin. Topological Groups, 2nd ed. New York: Gordon & Breach,

1966. 23. M. Reed and B. Simon. Methods of Modern Mathematical Physics, Vol. 1. New

York: Academic Press, 1972. 24. M. Reed and B. Simon. Methods of Modern Mathematical Physics, Vol. 2. New

York: Academic Press, 1975. 25. F. Riesz and B. Nagy. Functional Analysis. New York: Ungar, 1955. 26. W. Rudin. Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill,

1976. 27. W. Rudin. Functional Analysis, New York: McGraw-Hill, 1973. 28. P. Halmos. Measure Theory, Princeton, N.J.-New York: Van Nostrand, 1950. 29. E. Hewitt and K. Ross. Abstract Harmonic Analysis, Vol. 1. Berlin-New York:

Springer-Verlag, 1963. 30. E. Hewitt and K. Ross. Abstract Harmonic Analysis, Vol. 2. Berlin-New York:

Springer-Verlag, 1970. 31. G. E. Shi1ov. Generalized Functions and Partial Differential Equations. New York:

Gordon & Breach, 1968. 32. R. Edwards. Functional Analysis. Theory and Applications. New York: Holt,

Rinehart and Winston, 1965.

Supplementary Literature

1 *. P. S. Aleksandrov. Einfiihrung in die M engenlehre und die Theorie der reellen Funktionen. Berlin: VEB deutscher Verlag der Wissenschaft, 1956.

2*. S. Banach. Theorie des Operations Lineaires, New York: Chelsea, 1955. 3*. N. Bourbaki. General Topology. Fundamental Structures, 4th ed. Reading, Mass.:

Addison-Wesley, 1966. 4*. N. Bourbaki. General Topology. Use of the Real Numbers in General Topology.

Reading, Mass.: Addison-Wesley, 1966. 5*. N. Wiener. The Fourier Integral and Certain of its Applications. New York:

Dover, 1959. 6*. N. Wiener. Nonlinear Problems in Random Theory. New York: Wiley, 1958.

Bibliography 341

7*. N. Wiener and R. E. A. C. Paley. Fourier Transforms in the Complex Domain. Providence, R.I.: Amer. Math. Soc., 1934.

8*. M. M. Day. NormedLinearSpaces. Berlin: Springer-Verlag, 1958. 9*. J. Dieudonne. Foundations of Modern Analysis. New York: Academic Press, 1960.

10*. J. L. Kelley. General Topology. Princeton, N.J.: Van Nostrand, 1955. II*. H. Lebesgue. Lecons sur !'Integration et Ia Recherche des Fonctions Primitil'es.

2nd ed. Paris: Gauthier-Villars, 1928. 12*. L. Loomis. An Introduction to Abstract Harmonic Analysis. Princeton, N.J.:

Van Nostrand, 1953. 13*. V. P. Maslov. Operational Methods. Moscow: Mir, 1976. 14*. S. G. Mikhlin. Lectures on Linear Integral Equations. New York: Gordon &

Breach, 1960. 15*. A. Pietsch. Nukleare Lokalkonvexe Riiume. Berlin: Akademie-Verlag, 1965. 16*. A. I. Plesner. Spectral Theory of Linear Operators, Vols. I, II. New York: Ungar,

1969. 17*. A. Robertson and W. Robertson. Topological Vector Spaces. New York:

Cambridge Univ. Press, 1964. 18*. A. A. Fraenkel and Y. Bar-Hillel. Foundations of Set Theory. Amsterdam:

North-Holland, 1958. 19*. P. Halmos. Finite-Dimensional Vector Spaces, 2nd ed. Princeton, N.J.: Van

Nostrand, 1958. 20*. G. E. Shilov and Fan Dyk Tin'. Integral, Measure, and Derivative on Linear

Spaces. Moscow: Nauka, 1967 (Russian).

Distribution of Literature by Chapters

Chapter I. [2], [6], [15], [17], [18], [22], [1 *], [3*], [9*], [10*], [18*], [19*].

Chapter II. [2], [3], [4], [18], [26], [28], [II*], [20*].

Chapter III. [1], [2], [7], [8], [9], [10], [II], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [25], [27], [31], [32], [2*], [4*], [8*], [13*], [14*], [15*], [17*].

Chapter IV. [2], [8], [9], [15], [18], [22], [24], [27], [29], [30], [5*], [7*], [12*].

Chapter V. [5], [14], [15], [27], [16*].

Index

Absolute continuity 33 of a signed measure 36

Alexandrov line 142 Algebra

irreducible 220 of sets 12

Angle 86 Anti-linearity 90 Approximation dimension 55

Axiom of choice (Zermelo's axiom) 5

Ball, unit 3 8 Base of neighborhoods of zero 50 Basis, Hamel 274

Canonical imbedding of objects in a category into a sum 146

Canonical projection of a product of elements in a category onto a factor 147

Cantor set 180 Cantor staircase 165 Category 8 Centered system of sets 180 Cesaro means 316 Character on a group 104 Closure of an operator 125 Codimension 45 Cokernel 60 Commutative diagram 9 Commuting operators 127 Compact set 54

Complemented subspace 177 Completeness of a metric space 6 Completeness of an orthonormal system of

vectors 88 Completion 6 Congruent pairs of subspaces 201 Conjugacy class of elements 204 Conjugate numbers 47 Continuous basis (coherent or overfilled

system) 203 Convergence

almost everywhere 23 in measure 23 uniform 23

Convex body 173 Convex hull 171 Convolution of functions 96, 100

on the line 168 Countable monotonicity

of a measure 14 of an outer measure 15

Countable topological basis 189 Cross-norms 49

uniform 49 Cube, N-dimensional 175 Cyclic vector 128

<>-algebra 12 b-ring 12 Diameter of a partition 28 Direct product of generalized functions 83 Direct sum of algebras 220

343

344

Disjoint union 12 Distribution, tempered 78 Doubly stochastic matrix 181 Dual category 10 Dual group 104

Enveloping algebra (associative hull) 146 £-net 54 £-perpendicular 177 Equation

heat conduction 217 integral, of the second kind with degenerate

kernel 184 Volterra 184

Equicontinuous family of operators 51 Equimeasurable functions !57 Equivalence 3

of collections of seminorms 41 of signed measures 36 of well-ordered countable sets 141

Equivalence class 4 Equivalent categories II Essential spectrum of an operator 229 Essential supremum 26 Extension of an operator 125 Extreme point 56 Extreme subset 180

Field of p-adic numbers 144 Formula

Plancherel 112 Poisson 115 Stone's 229

Fourier coefficients I 07 Fourier transforms I 07

in the space S'(R) 114 of generalized functions I 09

Fredholm alternative 66 Fredholm operator criterion 63 Free

abelian group 145 group 145 Lie algebra 146

Function absolutely continuous 168 almost periodic 198 Bessel 208, 324 Borel 157 characteristic 13 (complex, vector-valued) m-measurable

22 continuous in the mean 186

Dirac (£~-function) 78, 79 Dirichlet 159 essentially bounded 26 Euler 141 generalized

Index

homogeneous, with degree of homo-geneity (Jc, s) 195

positive-definite 213 regular 78 with compact support 78

generating I 07 Haar 199 Heaviside 163 integrable 26 jump 163 measurable (11-measurable) 22, !58 Mobius 140

classical 234 positive-definite 210 quasi-periodic 216 Rademacher 199 simple 24 Steklov 212 test 78 Walsh 199 weakly measurable 176

Functional operator calculus 116 Functor, covariant (contravariant) 10

Generalized (nonunitary) character 209 Gibbs phenomenon 317 Graph of an operator 124 Group algebra 96

Half-ring 12 Hausdorff criterion 54 Heine-Bore! property 71 Hermitian

conjugate matrix 91 dual space 90 form 92 symmetry 85

Hilbert basis 89 Hilbert dimension 89 Homotopically equivalent sets 279 Hyperplane 45

Idempotent 220 Identity

Cayley's 325 Euler 182

Index

Hilbert 333 Sokhotskii 193

Index of an operator 60 Indicatrix, Banach 165 Inductive limit 148 Inequality

Bessel's 88 Cauchy-Bunyakovskii 85 Holder's 47 Minkowski integral 185

Integer lattice 96 Integer, p-adic 144 Integral

Fourier 107, 110 Lebesgue 27

of a simple function 25 Riemann-Stieltjes 29 Riemann, with respect to a projection­

valued measure 227 weak 176 with respect to a projection-valued measure

132 Integral sum

Lebesgue 132 Riemann 227 Riemann-Stieltjes 28

Interlacing principle 202 Inverse Fourier transformation I 09 Isomorphic objects of a category 8

Jordan block 220

Kernel 59

Largest (smallest) element 4 Lemma

Fatou's 32 Zorn's 5

Length of a word 241 Lexicographical order 141 Limit

of a sequence of sets 151 supremum (infimum) of a sequence of sets

151 Linear manifold 45 Luzin C-property 157

Majorant (minorant) 4 Mapping

bounded 189'

sequentially continuous 189

in the variables jointly 190 Maximal (minimal) element 4 Measure 13

countably additive (a-additive) 13 inner 152 Lebesgue 18 outer 14 projection-valued 131

345

quasi-invariant, with respect to translations 167

a-finite 17 Wiener 22 with a countable base 185

Measures, disjoint 17 Minkowski functional 39 Mixed area of a pair of sets 273 Mixed volume of a pair of sets 273 Morphism

functorial (natural transformation of func­tors) 10

identity 8 of a category 8

Natural domain 126 Net (direction) 4 Newton's formulas 326 Norm 38

of an operator 43 p-adic 143

Object of a category 8 One-parameter group 134 Opening between subspaces 201 Operator

adjoint 43 almost invertible 62 closed 125 compact (completely continuous) 57 continuous linear 50 creation (annihilation) 183, 213 essentially selfadjoint 125 Fredholm 60 Hermitian adjoint Hilbert-Schmidt Laplace 217 normal 91 nuclear 203

91 184, 202

occupation number 213 of finite rank 57 of fractional integration (fractional differ­

entiation) 208

346

Operator (collt.) particle number 213 positive 91 regular 219 selfadjoint (Hermitian) 91, 125 unbounded 124 unitary 91 Volterra 222 with simple spectrum 128

Order of a generalized function 80, 81

Orthogonal complement 86 projection 91 reflection 200

Orthogonality 86 Orthogonalization process 89 Orthonormal system of vectors 88

Parallelogram law 87, 199 Parseval's equality 88 Partial isometry 202 Partition of unity 190 Point

Lebesgue 157 of density of a set 154

Polar decomposition of an operator 202 Polynomials

Bernoulli 198 Legendre, Tchebycheff, Laguerre, Hermite

303 Pontryagin duality principle 105 Positive linear functional 187 Positive semidefiniteness 85 Principal value of an integral in the Cauchy

sense 192 Product

of objects in a category 147 of partially ordered sets 140 torsion (of groups) 147

Projective limit 148

Quadratic form 92 Quaternions 84 Quotient set 4

Range of an operator 59 Reflexivity 3 Region 69 Relation 3

equivalence 3 partial-order 4 total order 4 transposed 3

Index

Relations, composition of 3 Representation of the algebra C(X) 228 Resolvent of an operator 117 Resolvent set 119 Reversing arrows II Ring

generated by a collection 13 of subsets 12

Scalar product 84 Segment (interval) 38 Semicontinuity of a measure !52 Seminorm 38 Sequence

convex 200 3-shaped 205 exact 60

at a term 60 positive-definite 212 rapidly converging 27 semi -exact 60

Set 5 absorbing 38 balanced 38 bounded above (below) 4

in an LTS 171 in a polynormed space 71

convex 38 directed 4 Egorov 158 Lebesgue (of a function) 22 Lebesgue measurable 15

with respect to a <T-finite measure 17 level 159 measurable, in the Caratheodory sense

154 .u-measurable 15 of <T-uniqueness of a measure !54 pre-compact 54 well-ordered 5

<T-algebra 12 <T-ring 12 Signed measure (complex measure) 17 Signum, p-adic 144 Similar matrices 220 Slowly increasing sequence (sequence of

moderate growth) 109 Space

'Banach 42 cohomology 60 Hausdorff (separated) 41 Hilbert 85 linear topological 39

Index

locally compact 96 locally convex 40 metrizable 41 normable 41 normed 41 of almost periodic continuous functions

198 of compactly supported infinitely differ­

entiable functions 71 of infinitely differentiable functions 70 of rapidly decreasing infinitely differenti-

able functions 72 polynormed 40 pre-Hilbert 84 reflexive 43 strong dual 42

Spectral decomposition of an operator 134 Spectral measure 134 Spectral radius of an operator 121 Spectrum of an operator 118 Square root of an operator 202 Subset

Borel 13, 151 cylindrical 20

Sum of objects in a category 146 Support

of a function 71 of a generalized function 79

Symmetry 3 System

Haar 199 Rademacher 199 Walsh 199

Tensor algebra 146 Tensor product 49, 147 Theorem

Arzeh1-Ascoli 57 Banach inverse mapping (open mapping)

53 Banach-Steinhaus 51 Courant 93 Egorov 24 Fubini 35

Hahn-Banach 44, 45 Helly's 44, 176

first 188 second 188

Hilbert 93 kernel 194 Krein-Mil'man 56 Lebesgue 15

dominated convergence 31 Luzin 159

347

monotone convergence (of B. Levi) 32 Nikol'skii 62 Paley-Wiener 113, 215 Radon-Nikodym 36 Riesz 63 Stone's 134 Tychonoff 313 von Neumann's ergodic 229 Weierstrass 77, 205 Zermelo's 5

Topology convex core 171 strong operator 50 uniform 50 weak 42 weak operator 50 weak-* 42

Torus, n-dimensional 96 Trace of an operator 203 Transitivity 3

Uniformly bounded family of operators 51 Universal repelling object 8

Vacuum vector 213 Variation

of a function 19 of a signed measure 17

Weyl criterion 229 Word 241