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?at, 3.1.2 (5)
equivalent multinorms, 5.3.1
formal duality, 2.3.15
equivalent seminorms, 5.3.3
Fourier coe?cient family, 6.3.15
estimate for the diameter of
Fourier–Plancherel transform,
a spherical layer, 6.2.1
10.11.15
Euler identity, 8.5.17
Fourier–Schwartz transform,
evaluation mapping, 10.3.4 (3)
10.11.19
everywhere-de?ned operator, 2.2.1
Fourier series, 6.3.16
everywhere dense set, 4.7.3 (3)
Fourier transform
exact sequence, 2.3.4
of a distribution, 10.11.19
exact sequence at a term, 2.3.4
Fourier transform of a function,
exclave, 8.2.9
10.11.3
expanding mapping, Ex. 4.14
Fourier transform relative to
extended function, 3.4.2
a basis, 6.3.15
extended real axis, 3.8.1
Fr?chet space, 5.5.2
e
extended reals, 3.8.1
Fredholm Alternative, 8.5.6
extension of an operator, 2.3.6
Fredholm index, 8.5.1
exterior of a set, 4.1.13
Fredholm operator, 8.5.1
exterior point, 4.1.13
Fredholm Theorem, 8.5.8
Extreme and Discrete Lemma,
frontier of a set, 4.1.13
3.6.4
from A into/to B, 1.1.1
extreme point, 3.6.1
Глоссарий 333

Fubini Theorem for distributions, general form of a weakly
10.10.5 (8) continuous functional,
10.3.10
Fubini Theorem for measures,
general position, Ex. 3.10
10.9.4 (6)
generalized derivative in the
full subalgebra, 11.1.5
Sobolev sense, 10.10.5 (4)
fully norming set, 8.1.1
Generalized Dini Theorem, 10.8.6
Function Comparison Lemma,
generalized function, 10.10.4
3.8.3
function of class C (m) , 10.9.9 Generalized Riesz–Schauder
Theorem, 8.4.10
function of compact support, 9.6.4
generalized sequence, 1.2.16
Function Recovery Lemma, 3.8.2
Generalized Weierstrass Theorem,
functor, 10.9.4 (4)
10.9.9
fundamental net, 4.5.2
germ, 8.1.14
fundamental sequence, 4.5.2
GNS-construction, 11.9.11
fundamentally summable family
GNS-Construction Theorem,
of vectors, 5.5.9 (7
11.9.10
gauge, 3.8.6 gradient mapping, 6.4.2
gauge function, 3.8.6 Gram–Schmidt orthogonalization
Gauge Theorem, 3.8.7 process, 6.3.14
-correspondence, 3.1.6 graph norm, 7.4.17
-hull, 3.1.11 Graph Norm Principle, 7.4.17
-set, 3.1.1 greatest element, 1.2.6
Gelfand–Dunford Theorem in greatest lower bound, 1.2.9
an operator setting, 8.2.3 Grothendieck Criterion, 8.3.11
Gelfand–Dunford Theorem Grothendieck Theorem, 8.3.9
in an algebraic setting, 11.3.2 ground ?eld, 2.1.3
Gelfand formula, 5.6.8 ground ring, 2.1.1
Gelfand–Mazur Theorem, 11.2.3 group algebra, 10.9.4 (7)
Gelfand–Na? ?mark–Segal group character, 10.11.1
construction, 11.9.11
Haar integral, 10.9.4 (1)
Gelfand Theorem, 7.2.2
Hahn–Banach Theorem, 3.5.3
Gelfand transform of an algebra,
Hahn–Banach Theorem
11.6.8
in analytical form, 3.5.4
Gelfand transform of an element,
Hahn–Banach Theorem
11.6.8
in geometric form, 3.8.12
Gelfand Transform Theorem,
Hahn–Banach Theorem
11.6.9
in subdi?erential form, 3.5.4
general form of a compact
Hamel basis, 2.2.9 (5)
operator in Hilbert space,
6.6.9 Hausdor? Completion Theorem,
4.5.12
general form of a linear functional
in Hilbert space, 6.4.2 Hausdor? Criterion, 4.6.7
Глоссарий
334

Hausdor? metric, Ex. 4.8 Ideal Correspondence Principle,
7.3.5
Hausdor? multinorm, 5.1.8
Ideal Hahn–Banach Theorem,
Hausdor? multinormed space,
7.5.9
5.1.8
ideally convex function, 7.5.4
Hausdor? space, 9.3.5
ideally convex set, 7.1.3
Hausdor? Theorem, 7.6.12
idempotent operator, 2.2.9 (4)
Hausdor? topology, 9.3.5
identical embedding, 1.1.3 (3)
H-closed space, Ex. 9.10
identity, 10.9.4
Heaviside function, 10.10.5 (4)
identity element, 11.1.1
Hellinger–Toeplitz Theorem, 6.5.3
identity mapping, 1.1.3 (3)
hermitian element, 11.7.1
identity relation, 1.1.3 (3)
hermitian form, 6.1.1
image, 1.1.2
hermitian operator, 6.5.1
image of a ?lterbase, 1.3.5 (1)
hermitian state, 11.9.8
image of a set, 1.1.3 (5)
Hilbert basis, 6.3.8
image of a topology, 9.2.12
Hilbert cube, 9.2.17 (2)
image topology, 9.2.12
Hilbert dimension, 6.3.13
Image Topology Theorem,
Hilbert identity, 5.6.19 9.2.11
Hilbert isomorphy, 6.3.17 imaginary part of a function,
Hilbert–Schmidt norm, Ex. 8.9 5.5.9 (4)
Hilbert–Schmidt operator, increasing mapping, 1.2.3 (5)
Ex. 8.9 independent measure, 10.9.4 (3)
Hilbert–Schmidt Theorem, 6.6.7 index, 8.5.1
Hilbert space, 6.1.7 indicator function, 3.4.8 (2)
Hilbert-space isomorphism, 6.3.17 indiscrete topology, 9.1.8 (3)
Hilbert sum, 6.1.10 (5) induced relation, 1.2.3 (1)
H?lder inequality, 5.5.9 (4)
o induced topology, 9.2.17 (1)
holey disk, 4.8.5 inductive limit topology, 10.9.6
holomorphic function, 8.1.4 inductive set, 1.2.19
in?mum, 1.2.9
Holomorphy Theorem, 8.1.5
in?nite-rank operator, 6.6.8
homeomorphism, 9.2.4
in?nite set, 5.5.9 (3)
homomorphism, 7.4.1
inner product, 6.1.4
H?rmander transform, Ex. 3.19
o
integrable function, 5.5.9 (4)
hyperplane, 3.8.9
integral, 5.5.9 (4)
hypersubspace, 3.8.
integral with respect to
ideal, 11.4.1 a measure, 10.9.3
Ideal and Character Theorem, interior of a set, 4.1.13
11.6.6 interior point, 4.1.13
ideal correspondence, 7.3.3 intersection of topologies, 9.1.14
interval, 3.2.15
Ideal Correspondence Lemma,
7.3.4 Interval Addition Lemma, 3.2.15
Глоссарий 335

invariant subspace, 2.2.9 (4) Jordan Curve Theorem, 4.8.3
juxtaposition, 2.2.
inverse-closed subalgebra, 11.1.5
inverse image of a multinorm,
Kakutani Criterion, 10.7.1
5.1.10 (3)
Kakutani Lemma, 10.8.7
inverse image of a preorder,
Kakutani Theorem, 7.4.11 (3)
1.2.3 (3)
Kantorovich space, 3.2.8
inverse image of a seminorm, 5.1.4
Kantorovich Theorem, 3.3.4
inverse image of a set, 1.1.3 (5)
Kaplansky–Fukamija Lemma,
inverse image of a topology, 9.2.9
11.9.7
inverse image of a uniformity,
Kato Criterion, 7.4.19
9.5.5 (3)
kernel of an operator, 2.3.1
inverse image topology, 9.2.9
ket-mapping, 10.3.1
Inverse Image Topology Theorem,
ket-topology, 10.3.5
9.2.8
Kolmogorov Normability
inverse of a correspondence,
Criterion, 5.4.5
1.1.3 (1)
Kre??n–Milman Theorem, 10.6.5
inverse of an element Kre??n–Milman Theorem
in an algebra, 11.1.5 in subdi?erential form, 3.6.5
Inversion Theorem, 10.11.12 Kre??n–Rutman Theorem, 3.3.5
invertible element, 11.1.5 Krull Theorem, 11.4.8
invertible operator, 5.6.10 Kuratowski–Zorn Lemma, 1.2.20
involution, 6.4.13 K-space, 3.2.8
involutive algebra, 6.4.13 K-ultrametric, 9.5.13
irreducible representation, 8.2.2
last element, 1.2.6
irre?exive space, 5.1.10 (8)
lattice, 1.2.12
isolated part of a spectrum, 8.2.9
lear trap map, 3.7.4
isolated point, 8.4.7
least element, 1.2.6
isometric embedding, 4.5.11
Lebesgue measure, 10.9.4 (1)
isometric isomorphism of algebras,
Lebesgue set, 3.8.1
11.1.8
Lefschetz Lemma, 9.6.3
isometric mapping, 4.5.11
left approximate inverse, 8.5.9
isometric representation, 11.1.8
left Haar measure, 10.9.4 (1)
isometric ?-isomorphism, 11.8.3
left inverse of an element
isometric ?-representation, 11.8.3
in an algebra, 11.1.3
isometry into, 4.5.11
lemma on continuity of a convex
isometry onto, 4.5.11
function, 7.5.1
isomorphism, 2.2.5
lemma on the numeric range
isotone mapping, 1.2
of a hermitian element,
James Theorem, 10.7.5 11.9.3
Jensen inequality, 3.4.5 level set, 3.8.1
join, 1.2.12 Levy Projection Theorem, 6.2.2
limit of a ?lterbase, 4.1.16
Jordan arc, 4.8.2
Глоссарий
336

Lindenstrauss space, 5.5.9 (5) maximal element, 1.2.10
maximal ideal, 11.4.5
Lindenstrauss–Tzafriri Theorem,
maximal ideal space, 11.6.7
7.4.11 (3)
Maximal Ideal Theorem, 11.5.3
linear change of a variable under
Mazur Theorem, 10.4.9
the subdi?erential sign, 3.5.4
meager set, 4.7.1
linear combination, 2.3.12
measure, 10.9.3
linear correspondence, 2.2.1,
Measure Localization Principle,
12; 3.1.7
10.9.10
linear functional, 2.2.4
measure space, 5.5.9 (4)
linear operator, 2.2.1
meet, 1.2.12
linear representation, 8.2.2
member of a set, 1.1.3 (4)
linear set, 2.1.4 (3)
metric, 4.1.1
linear space, 2.1.4 (3)
metric space, 4.1.1
linear span, 3.1.14
metric topology, 4.1.9
linear topological space, 10.1.3
metric uniformity, 4.1.5
linear topology, 10.1.3
Metrizability Criterion, 5.4.2
linearly independent set, 2.2.9 (5)
metrizable multinormed space,
linearly-ordered set, 1.2.19
5.4.1
Lions Theorem of Supports,
minimal element, 1.2.10
10.10.5 (9)
Minimal Ideal Theorem, 11.5.1
Liouville Theorem, 8.1.10
Minkowski–Ascoli–Mazur
local data, 10.9.11
Theorem, 3.8.12
locally compact group, 10.9.4 (1)
Minkowski functional, 3.8.6
locally compact space, 9.4.20
Minkowski inequality, 5.5.9 (4)
locally compact topology, 9.4.20
minorizing set, 3.3.2
locally convex space, 10.2.9
mirror, 10.2.7
locally convex topology, 10.2.1
module, 2.1.1
locally ?nite cover, 9.6.2
modulus of a scalar, 5.1.10 (4)
locally integrable function, 9.6.17
modulus of a vector, 3.2.12
locally Lipschitz function, 7.5.6
molli?er, 9.6.14
loop, 4.8.2
mollifying kernel, 9.6.14
lower bound, 1.2.4
monomorphism, 2.3.1
lower limit, 4.3.5
monoquotient, 2.3.11
lower right Dini derivative, 4.7.7
Montel space, 10.10.9 (2)
lower semicontinuous, 4.3.3
Moore subnet, 1.3.5 (2)
L2 -Fourier transform, 10.11.15
morphism, 8.2.2, 126; 11.1.2
Mackey–Arens Theorem, 10.4.5 morphism representing
Mackey Theorem, 10.4.6 an algebra, 8.2.2
Mackey topology, 10.4.4 Motzkin formula, 3.1.13 (5)
mapping, 1.1.3 (3) multimetric, 9.5.9
massive subspace, 3.3.2 multimetric space, 9.5.9
matrix form, 2.2.9 (4) multimetric uniformity, 9.5.9
Глоссарий 337

multimetrizable topological non-everywhere-de?ned operator,
space, 9.5.10 2.2.1
nonmeager set, 4.7.1
multimetrizable uniform space,
9.5.10 nonpointed cone, 3.1.2 (4)
multinorm, 5.1.6 nonre?exive space, 5.1.10 (8)
Multinorm Comparison Theorem, norm, 5.1.9
5.3.2 norm convergence, 5.5.9 (7)
multinorm summable family normable multinormed space,
of vectors, 5.5.9 (7) 5.4.1
multinormed space, 5.1.6 normal element, 11.7.1
multiplication formula, 10.11.5 normal operator, Ex. 8.17
multiplication of a germ normal space, 9.3.11
by a complex number, 8.1.16 normalized element, 6.3.5
multiplicative linear operator, normally solvable operator, 7.6.9
8.2.2 normative inequality, 5.1.10 (7)
normed algebra, 5.6.3
natural order, 3.2.6 (1)
normed dual, 5.1.10 (8)
negative part, 3.2.12 normed space, 5.1.9
neighborhood about a point, normed space of bounded
9.1.1 (2) elements, 5.5.9 (5)
neighborhood about a point norming set, 8.1.1
in a metric space, 4.1.9
norm-one element, 5.5.6
neighborhood ?lter, 4.1.10 nowhere dense set, 4.7.1
neighborhood ?lter of a set, 9.3.7 nullity, 8.5.1
neighborhood of a set, 8.1.13 (2), numeric family, 1.1.3 (4)
124; 9.3.7
numeric function, 9.6.4
Nested Ball Theorem, 4.5.7
numeric range, 11.9.1
nested sequence, 4.5.7 numeric set, 1.1.3 (
net, 1.2.16
net having a subnet, 1.3.5 (2) one-point compacti?cation, 9.4.22
net lacking a subnet, 1.3.5 (2) one-to-one correspondence,
1.1.3 (3)
Neumann series, 5.6.9
open ball, 4.1.3
Neumann Series Expansion
open ball of RN , 9.6.16
Theorem, 5.6.9
open correspondence, 7.3.12
neutral element, 2.1.4 (3), 11;
10.9.4 Open Correspondence Principle,
7.3.13
Nikol ski? Criterion, 8.5.22
?
open cylinder, 4.1.3
Noether Criterion, 8.5.14
open half-space, Ex. 3.3
nonarchimedean element,
Open Mapping Theorem, 7.4.6
5.5.9 (5)
open segment, 3.6.1
nonconvex cone, 3.1.2 (4)
open set, 9.1.4
Nonempty Subdi?erential
Theorem, 3.5.8 open set in a metric space, 4.1.11
Глоссарий
338

openness at a point, 7.3.6 partial operator, 2.2.1
partial order, 1.2.2
operator, 2.2.1
partial sum, 5.5.9 (7)
operator ideal, 8.3.3
partition of unity, 9.6.6
operator norm, 5.1.10 (7)
partition of unity subordinate

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