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f u, 10.10.5 (7)
coker T , 2.3.1
fn f , 10.10.7 (3)
core U , 3.4.11
fn 0, 10.9.8
diam U , 4.5.3
K
dim X, 2.2.9 (5)
g, 10.11.2
dom f , 3.4.2
?
dom F , 1.1.2 g(f ), 8.2.6
epi f , 3.4.2 h , 6.3.5
ext V , 3.6.1 lp , lp (E), 5.5.9 (4)
?l B, 1.3.3 l? , l? (E), 5.5.9 (2)
fr U , 4.1.13 m, 5.5.9 (2)
im F , 1.1.2 p q, 5.3.3
inf U , 1.2.9 pe , 5.5.9 (5)
int U , 4.1.13 pS , 3.8.6
ker T , 2.3.1 p T , 5.1.3
lin(U ), 3.1.14 pX/X0 , 5.1.10 (5)
pol , 10.5.7 r(T ), 5.6.6
rank T , 8.5.7 (2) s, Упр. 1.19
t? , 10.11.5 (8)
res(a), 11.2.1
res(T ), 5.6.13 ug , 10.10.5 (1)
u? , 10.10.5 (5)
seg, 3.6.1
sup U , 1.2.9 u ? f , 10.10.5 (9)
u1 ? u2 , 10.10.5 (8)
supp(f ), 9.6.4
u1 ? u2 , 10.10.5 (8)
supp(µ), 10.8.11, 10.9.4 (5)
x |, 10.3.1
supp(u), 10.10.5 (6)
a, 11.6.8 x , 6.4.1
aµ, 10.8.15 x , 5.1.10 (8)
x? , 10.11.5 (8)
a ? f , 10.9.4 (1)
(a, b)s , 11.9.9 x+ , 3.2.12
Указатель обозначений
326

· ? , 5.5.9 (5)
x? , 3.2.12
|x|, 3.2.12 · X , 5.1.9
· |X , 5.1.9
x p , 5.5.9 (4)
1, 5.3.10, 10.8.4 (6)
x ? , 5.5.9 (2)
2X , 1.2.3 (4)
? (x), 10.11.4
?X0 , 2.1.4 (6) ?, 6.4.13
x := x , 5.5.9 (7)
e?E e , 10.9.1
x > x , 6.4.1 , 5.5.9 (6)
E
x1 ? x2 , x1 ? x2 , 1.2.12 | · , 10.3.1
(x1 , x2 ), 1.2.12
· | · , 10.3.1
x | f , 5.1.11
· |, 10.3.1
x ?? y, 1.2.2
?, 1.2.2
x ? y, 5.5.6
X? , 2.1.4 (5)
x ? y, 6.2.5 ??
| y , 10.3.1 X? , 2.1.4 (4)
??
|||y|||p , 5.5.9 (6)
h(z)R(z)dz, 8.1.20
· , 5.1.9
· n,Q , 10.10.2 (2) , 11.6.8
Глоссарий



algebra of germs of holomorphic
Absolute Bipolar Theorem, 10.5.9
functions, 8.1.18
absolute concept, 9.4.7
algebraic basis, 2.2.9 (5)
absolute polar, 10.5.7
algebraic complement, 2.1.7
absolutely continuous measure,
algebraic dual, 2.2.4
10.9.4 (3)
algebraic isomorphism, 2.2.5
absolutely convex set, 3.1.2 (6)
algebraic subdi?erential, 7.5.8
absolutely fundamental family
of vectors, 5.5.9 (7) algebraically complementary
subspace, 2.1.7
absorbing set, 3.4.9
algebraically interior point, 3.4.11
addition in a vector space, 2.1.3
algebraically isomorphic spaces,
adherence of a ?lterbase, 9.4.1
2.2.6
adherent point, 4.1.13
algebraically re?exive space,
adherent point of a ?lterbase,
Ex. 2.8
9.4.1
ambient space, 2.1.4 (3)
adjoint diagram, 6.4.8
annihilator, 7.6.8
adjoint of an operator, 6.4.5
antidiscrete topology, 9.1.8 (3)
adjunction of unity, 11.1.2,
antisymmetric relation, 1.2.1
a?ne hull, 3.1.14
antitone mapping, 1.2.3
a?ne mapping, 3.1.7,
approximate inverse, 8.5.9
a?ne operator, 3.4.8 (4)
approximately invertible operator,
a?ne variety, 3.1.2 (5)
8.5.9
agreement condition, 10.9.4 (4)
approximation property, 8.3.10
Akilov Criterion, 10.5.3
approximation property in Hilbert
Alaoglu–Bourbaki Theorem,
space, 6.6.10
10.6.7
arc, 4.8.2
Alexandro? compacti?cation,
Arens multinorm, 8.3.8
9.4.22
ascent, Ex. 8.10
algebra, 5.6.2
Ascoli–Arzel` Theorem, 4.6.10
a
algebra of bounded operators,
assignment operator
5.6.5
Глоссарий
328

associate seminorm, 6.1.7 Banach’s Fundamental Principle,
7.1.5
associated Hausdor? pre-Hilbert
Banach’s Fundamental Principle
space, 6.1.10 (4)
for a Correspondence, 7.3.7
associated Hilbert space,
Banach–Steinhaus Theorem, 7.2.9
6.1.10 (4)
barrel, 10.10.9 (1)
associated multinormed space,
barreled normed space, 7.1.8
10.2.7
barreled space, 10.10.9 (1)
associated topology, 9.1.12
base for a ?lter, 1.3.3
associativity of least upper
basic ?eld, 2.1.2
bounds, 3.2.10
Bessel inequality, 6.3.7
asymmetric balanced
best approximation, 6.2.3
Hahn–Banach formula, 3.7.10
Beurling–Gelfand formula,
asymmetric Hahn–Banach
8.1.12 (2)
formula, 3.5.5
bilateral ideal, 8.3.3, 132; 11.6.2
Atkinson Theorem, 8.5.18
bilinear form, 6.1.2
Automatic Continuity Principle,
bipolar, 10.5.5
7.5.5
Bipolar Theorem, 10.5.8
automorphism, 10.11.4
Birkho? Theorem, 9.2.2
Bochner integral, 5.5.9 (6)
Baire Category Theorem, 4.7.6
bornological space, 10.10.9 (3)
Baire space, 4.7.2
boundary of an algebra, Ex. 11.8
Balanced Hahn–Banach Theorem,
boundary of a set, 4.1.13
3.7.13
boundary point, 4.1.13
Balanced Hahn–Banach Theorem
bounded above, 1.2.19
in a topological setting,
bounded below, 3.2.9
7.5.10
bounded endomorphism algebra,
balanced set, 3.1.2 (7)
5.6.5
balanced subdi?erential, 3.7.8
Bounded Index Stability
Balanced Subdi?erential Lemma, Theorem, 8.5.21
3.7.9
bounded operator, 5.1.10 (7)
ball, 9.6.14 bounded Radon measure,
Banach algebra, 5.6.3 10.9.4 (2)
Banach Closed Graph Theorem, bounded set, 5.4.3
7.4.7 boundedly order complete
Banach Homomorphism Theorem, lattice, 3.2.8
7.4.4 Bourbaki Criterion, 4.4.7,
Banach Inversion Stability 46; 9.4.4
Theorem, 5.6.12 bracketing of vector spaces, 10.3.1
Banach Isomorphism Theorem, bra-functional, 10.3.1
7.4.5 bra-mapping, 10.3.1
Banach range, 7.4.18 bra-topology, 10.3.5
B-stable, 10.1.8
Banach space, 5.5.1
Глоссарий 329

bump function, 9.6.19 closure operator, Ex. 1.11
coarser cover, 9.6.1
Calkin algebra, 8.3.5 coarser ?lter, 1.3.6
Calkin Theorem, 8.3.4 coarser pretopology, 9.1.2
canonical embedding, 5.1.10 (8) codimension, 2.2.9 (5)
canonical exact sequence, codomain, 1.1.2
2.3.5 (6) co?nite set, Ex. 1.19
canonical operator representation, coimage of an operator, 2.3.1
11.1.7 coincidence of the algebraic and
canonical projection, 1.2.3 (4) topological subdi?erentials,
Cantor Criterion, 4.5.6 7.5.8
Cantor Theorem, 4.4.9 coinitial set, 3.3.2
cap, 3.6.3 (4) cokernel of an operator, 2.3.1
Cauchy–Bunyakovski? ?–Schwarz comeager set, 4.7.4
inequality, 6.1.5 commutative diagram, 2.3.3
Cauchy ?lter, 4.5.2 Commutative Gelfand–Na? ?mark
Cauchy net, 4.5.2 Theorem, 11.8.4
Cauchy–Wiener Integral Theorem, compact convergence, 7.2.10
8.1.7 Compact Index Stability
centralizer, 11.1.6 Theorem, 8.5.20
chain, 1.2.19 compact-open topology, 8.3.8
character group, 10.11.2 compact operator, 6.6.1
character of a group algebra, compact set, 9.4.2
10.11.1 (1) compact set in a metric space,
character of an algebra, 11.6.4 4.4.1
character space of an algebra, compact space, 9.4.7
11.6.4 compact topology, 9.4.7
characteristic function, 5.5.9 (6) compactly-supported distribution,
charge, 10.9.4 (3) 10.10.5 (6)
Chebyshev metric, 4.6.8 compactly-supported function,
classical Banach space, 5.5.9 (5) 9.6.4
clopen part of a spectrum, 8.2.9 compactum, 9.4.17
closed ball, 4.1.3 compatible topology, 10.4.1
closed convex hull, 10.6.5 complementary projection,
closed correspondence, 7.3.8 2.2.9 (4)
closed cylinder, 4.1.3 complementary subspace, 7.4.9
closed-graph correspondence, 7.3.9 Complementation Principle, 7.4.10
closed half-space, Ex. 3.3 complemented subspace, 7.4.9
closed linear span, 10.5.6 complement of an orthoprojection,
closed set, 9.1.4 6.2.10
closed set in a metric space, complement of a projection,
4.1.11 2.2.9 (4)
closure of a set, 4.1.13 complete lattice, 1.2.13
Глоссарий
330

complete metric space, 4.5.5 contour integral, 8.1.20
complete set, 4.5.14 conventional summation, 5.5.9 (4)
completely regular space, 9.3.15 convergent ?lterbase, 4.1.16
completion, 4.5.13 convergent net, 4.1.17
complex conjugate, 2.1.4 (2) convergent sequence space,
complex distribution, 10.10.5 (5) 3.3.1 (2)
complex plane, 8.1.3 convex combination, 3.1.14
complex vector space, 2.1.3 convex correspondence, 3.1.7
complexi?cation, 8.4.8 convex function, 3.4.4
complexi?er, 3.7.4 convex hull, 3.1.14
composite correspondence, 1.1.4 convex set, 3.1.2 (8)
Composite Function Theorem, convolution algebra, 10.9.4 (7)
8.2.8 convolution of a measure and
composition, 1.1.4 a function, 10.9.4 (7)
Composition Spectrum Theorem, convolution of distributions,
5.6.22 10.10.5 (9)
cone, 3.1.2 (4) convolution of functions, 9.6.17
conical hull, 3.1.14 convolution of measures,
conical segment, 3.1.2 (9) 10.9.4 (7)
conical slice, 3.1.2 (9) convolutive distributions,
conjugate distribution, 10.10.5 (5) 10.10.5 (9)
conjugate exponent, 5.5.9 (4) coordinate projection, 2.2.9 (3)
conjugate-linear functional, 2.2.4 coordinatewise operation,
conjugate measure, 10.9.4 (3) 2.1.4 (4)
connected elementary compactum, core, 3.4.11
4.8.5 correspondence, 1.1.1
connected set, 4.8.4 correspondence in two arguments,
constant function, 5.3.10, 64; 1.1.3 (6)
10.8.4 (6) correspondence onto, 1.1.3 (3)
Continuous Extension Principle, coset, 1.2.3 (4)
7.5.11 coset mapping, 1.2.3 (4)
continuous function at a point, countable convex combination,
4.2.2, 43; 9.2.5 7.1.3
Continuous Function Recovery Countable Partition Theorem,
Lemma, 9.3.12 9.6.20
continuous functional calculus, countable sequence, 1.2.16
11.8.7 countably normable space, 5.4.1
continuous mapping of a metric cover of a set, 9.6.1
space, 4.2.2 C ? -algebra, 6.4.13
continuous mapping C ? -subalgebra, 11.7.8
of a topological space, 9.2.4
Davis–Figiel–Szankowski
continuous partition of unity,
Counterexample, 8.3.14
9.6.6
Глоссарий 331

de Branges Lemma, 10.8.16 distance, 4.1.1
decomplexi?cation, 6.1.10 (2) distribution, 10.10.4
decomposition reduces distribution applies to a function,
an operator, 2.2.9 (4) 10.10.5 (7)
decreasing mapping, 1.2.3 Distribution Localization
Dedekind complete vector Principle, 10.10.12
lattice, 3.2.8 distribution of ?nite order,
de?ciency, 8.5.1 10.10.5 (3)
delta-function, 10.9.4 (1) distribution size at most m,
delta-like sequence, 9.6.15 10.10.5 (3)
?-like sequence, 9.6.15 distribution of slow growth,
?-sequence, 9.6.15 10.11.16
dense set, 4.5.10 distributions admitting
denseness, 4.5.10 convolution, 10.10.5 (9)
density of a measure, 10.9.4 (3) distributions convolute, 10.10.5 (9)
derivative in the distribution division algebra, 11.2.3
sense, 10.10.5 (4)
domain, 1.1.2
derivative of a distribution,
Dominated Extension Theorem,
10.10.5 (4)
3.5.4
descent, Ex. 8.10
Double Prime Lemma, 7.6.6
diagonal, 1.1.3 (3)
double prime mapping, 5.1.10 (8)
diagram prime, 7.6.5
double sharp, Ex. 2.7
Diagram Prime Principle, 7.6.7
downward-?ltered set, 1.2.15
diagram star, 6.4.8
dual diagram, 7.6.5
Diagram Star Principle, 6.4.9
dual group, 10.11.2
diameter, 4.5.3
dual norm of a functional,
Diedonn` Lemma, 9.4.18
e
5.1.10 (8)
dimension, 2.2.9 (5)
dual of a locally convex space,
Dini Theorem, 7.2.10
10.2.11
Dirac measure, 10.9.4 (1)
dual of an operator, 7.6.2
direct polar, 7.6.8, 116; 10.5.1
duality bracket, 10.3.3
direct sum decomposition, 2.1.7
duality pair, 10.3.3
direct sum of vector spaces,
2.1.4 (5) dualization, 10.3.3
directed set, 1.2.15 Dualization Theorem, 10.3.9
direction, 1.2.15 Dunford–Hille Theorem, 8.1.3
directional derivative, 3.4.12 Dunford Theorem, 8.2.7 (2)
discrete element, 3.3.6 Dvoretzky–Rogers Theorem,
Discrete Kre? ?n–Rutman Theorem, 5.5.9 (7)
3.3.8 dyadic-rational point, 9.3.13
discrete topology, 9.1.8 (4)
disjoint measures, 10.9.4 (3) e?ective domain of de?nition,
disjoint sets, 4.1.10 3.4.2
Глоссарий
332

extreme set, 3.6.
Eidelheit Separation Theorem,
3.8.14
face, 3.6.1
eigenvalue, 6.6.3 (4)
factor set, 1.2.3 (4)
eigenvector, 6.6.3
faithful representation, 8.2.2
element of a set, 1.1.3 (4)
family, 1.1.3 (4)
elementary compactum, 4.8.5
?lter, 1.3.3
endomorphism, 2.2.1, 12; 8.2.1
?lterbase, 1.3.1
endomorphism algebra, 2.2.8,
?ner cover, 9.6.1
13; 5.6.5
?ner ?lter, 1.3.6
endomorphism space, 2.2.8
?ner multinorm, 5.3.1
En?o counterexample, 8.3.12
?ner pretopology, 9.1.2
entourage, 4.1.5
?ner seminorm, 5.3.3
envelope, Ex. 1.11
?nest multinorm, 5.1.10 (2)
epigraph, 3.4.2
?nite complement ?lter, 5.5.9 (3)
epimorphism, 2.3.1
?nite descent, Ex. 8.10
?-net, 8.3.2
?-perpendicular, 8.4.1 ?nite-rank operator, 6.6.8,
?-Perpendicular Lemma, 8.4.1 97; 8.3.6
?nite-valued function, 5.5.9 (6)
Equicontinuity Principle, 7.2.4
?rst category set, 4.7.1
equicontinuous set, 4.2.8
?rst element, 1.2.6
equivalence, 1.2.2
?xed point, Ex. 1.11
equivalence class, 1.2.3 (4)

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