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operator representation, 8.2.2
to a cover, 9.6.7
order, 1.2.2
patch, 10.9.11
order by inclusion, 1.3.1
perforated disk, 4.8.5
order compatible with vector
periodic distribution, 10.11.17 (7)
structure, 3.2.1
Pettis Theorem, 10.7.4
order ideal, 10.8.11
Phillips Theorem, 7.4.13
order of a distribution, 10.10.5 (3)
Plancherel Theorem, 10.11.14
ordered set, 1.2.2
point ?nite cover, 9.6.2
ordered vector space, 3.2.1
point in a metric space, 4.1.1
ordering, 1.2.2
point in a space, 2.1.4 (3)
ordering cone, 3.2.4
point in a vector space, 2.1.3
oriented envelope, 4.8.8
pointwise convergence, 9.5.5 (6)
orthocomplement, 6.2.5
pointwise operation, 2.1.4 (4)
orthogonal complement, 6.2.5
polar, 7.6.8, 116; 10.5.1
orthogonal family, 6.3.1
Polar Lemma, 7.6.11
orthogonal orthoprojections,
polarization identity, 6.1.3
6.2.12
Pontryagin–van Kampen Duality
orthogonal set, 6.3.1
Theorem, 10.11.2
orthogonal vectors, 6.2.5
poset, 1.2.2
orthonormal family, 6.3.6
positive cone, 3.2.5
orthonormal set, 6.3.6
positive de?nite inner product,
orthonormalized family, 6.3.6
6.1.4
orthoprojection, 6.2.7
positive distribution, 10.10.5 (2)
Orthoprojection Summation
positive element of a C ? -algebra,
Theorem, 6.3.3
11.9.4
Orthoprojection Theorem, 6.2.10
positive form on a C ? -algebra,
Osgood Theorem, 4.7.
Ex. 11.11
pair-dual space, 10.3.3 positive hermitian form, 6.1.4
pairing, 10.3.3 positive matrix, Ex. 3.13
pairwise orthogonality of ?nitely positive operator, 3.2.6 (3)
many orthoprojections, positive part, 3.2.12
6.2.14 positive semide?nite hermitian
paracompact space, 9.6.9 form, 6.1.4
Parallelogram Law, 6.1.8 positively homogeneous
Parseval identity, 6.3.16, 89; functional, 3.4.7 (2)
10.11.12 powerset, 1.2.3 (4)
part of an operator, 2.2.9 (4) precompact set, Ex. 9.16
partial correspondence, 1.1.3 (6) pre-Hilbert space, 6.1.7
339

preimage of a multinorm, quotient of a seminormed space,
5.1.10 (3) 5.1.10 (5)
quotient seminorm, 5.1.10 (5)
preimage of a seminorm, 5.1.4
quotient set, 1.2.3 (4)
preimage of a set, 1.1.3 (5)
quotient space of a multinormed
preintegral, 5.5.9 (4)
space, 5.3.11
preneighborhood, 9.1.1 (2)
quotient vector space, 2.1.4 (6
preorder, 1.2.2
preordered set, 1.2.2
radical, 11.6.11
preordered vector space, 3.2.1
Radon F-measure, 10.9.3
presheaf, 10.9.4 (4)
Radon–Nikod?m Theorem,
y
pretopological space, 9.1.1 (2)
10.9.4 (3)
pretopology, 9.1.1
range of a correspondence, 1.1.2
primary Banach space, Ex. 7.17
rank, 8.5.7 (2)
prime mapping, 6.4.1
rare set, 4.7.1
Prime Theorem, 10.2.13
Rayleigh Theorem, 6.5.2
Principal Theorem of the
real axis, 2.1.2
Holomorphic Functional
real carrier, 3.7.1
Calculus, 8.2.4
real C-measure, 10.9.4 (3)
product, 4.3.2
real distribution, 10.10.5 (5)
product of a distribution and
real hyperplane, 3.8.9
a function, 10.10.5 (7)
real measure, 10.9.4
product of germs, 8.1.16
real part map, 3.7.2
product of sets, 1.1.1, 1; 2.1.4 (4)
real part of a function, 5.5.9 (4)
product of topologies, 9.2.17 (2)
real part of a number, 2.1.2
product of vector spaces, 2.1.4 (4)
real subspace, 3.1.2 (3)
product topology, 4.3.2, 44;
real vector space, 2.1.3
9.2.17 (2)
reali?cation, 3.7.1
projection onto X1 along X2 ,
reali?cation of a pre-Hilbert
2.2.9 (4)
space, 6.1.10 (2)
projection to a set, 6.2.3
reali?er, 3.7.2
proper ideal, 11.4.5
reducible representation, 8.2.2
pseudometric, 9.5.7
re?nement, 9.6.1
p-sum, 5.5.9 (6)
re?ection of a function, 10.10.5
p-summable family, 5.5.9 (4)
re?exive relation, 1.2.1
punctured compactum, 9.4.21
re?exive space, 5.1.10 (8)
pure subalgebra, 11.1.5
regular distribution, 10.10.5 (1)
Pythagoras Lemma, 6.2.8
regular operator, 3.2.6 (3)
Pythagoras Theorem, 6.3.
regular space, 9.3.9
quasinilpotent, Ex. 8.18 regular value of an operator,
quotient mapping, 1.2.3 (4) 5.6.13
quotient multinorm, 5.3.11 relation, 1.1.3 (2)
quotient of a mapping, 1.2.3 (4) relative topology, 9.2.17 (1)

340

relatively compact set, 4.4.4 Riesz idempotent, 8.2.11
removable singularity, 8.2.5 (2) Riesz–Kantorovich Theorem,
3.2.17
representation, 8.2.2
representation space, 8.2.2 Riesz operator, Ex. 8.15
reproducing cone, Ex. 7.12 Riesz Prime Theorem, 6.4.1
residual set, 4.7.4 Riesz projection, 8.2.11
resolvent of an element Riesz–Schauder operator,
of an algebra, 11.2.1 Ex. 8.11
resolvent of an operator, 5.6.13 Riesz–Schauder Theorem, 8.4.8
resolvent set of an operator, Riesz space, 3.2.7
5.6.13 Riesz Theorem, 5.3.5
resolvent value of an element right approximate inverse, 8.5.9
of an algebra, 11.2.1 right Haar measure, 10.9.4 (1)
resolvent value of an operator, right inverse of an element
5.6.13 in an algebra, 11.1.3
R-measure, 10.9.4 (3)
restriction, 1.1.3 (5)
restriction of a distribution, rough draft, 4.8.8
10.10.5 (6) row-by-column rule, 2.2.9 (4
restriction of a measure,
salient cone, 3.2.4
10.9.4 (4)
Sard Theorem, 7.4.12
restriction operator, 10.9.4 (4)
scalar, 2.1.3
reversal, 1.2.5
scalar ?eld, 2.1.3
reverse order, 1.2.3 (2)
scalar multiplication, 2.1.3
reverse polar, 7.6.8, 116; 10.5.1
scalar product, 6.1.4
reversed multiplication, 11.1.6
scalar-valued function, 9.6.4
Riemann function, 4.7.7
Schauder Theorem, 8.4.6
Riemann–Lebesgue Lemma,
Schwartz space of distributions,
10.11.5 (3)
10.11.16
Riemann Theorem on Series,
Schwartz space of functions,
5.5.9 (7)
10.11.6
Riesz Criterion, 8.4.2
Schwartz Theorem, 10.10.10
Riesz Decomposition Property,
second dual, 5.1.10 (8)
3.2.16
selfadjoint operator, 6.5.1
Riesz–Dunford integral, 8.2.1
semi-extended real axis, 3.4.1
Riesz–Dunford Integral
semi-Fredholm operator,
Decomposition Theorem,
Ex. 8.13
8.2.13
semi-inner product, 6.1.4
Riesz–Dunford integral
semimetric, 9.5.7
in an algebraic setting, 11.3.1
semimetric space, 9.5.7
Riesz–Fisher Completeness
seminorm, 3.7.6
Theorem, 5.5.9 (4)
Riesz–Fisher Isomorphism seminorm associated with
Theorem, 6.3.16 a positive element, 5.5.9 (5)
341

seminormable space, 5.4.6 simple function, 5.5.9 (6)
seminormed space, 5.1.5 simple Jordan loop, 4.8.2
semisimple algebra, 11.6.11 single-valued correspondence,
separable space, 6.3.14 1.1.3 (3)
separated multinorm, 5.1.8 Singularity Condensation
Principle, 7.2.12
separated multinormed space,
5.1.8 Singularity Fixation Principle,
separated topological space, 9.3.2 7.2.11
separated topology, 9.3.2 skew ?eld, 11.2.3
separating hyperplane, 3.8.13 slowly increasing distribution,
Separation Theorem, 3.8.11 10.11.16
Sequence Prime Principle, 7.6.13 smooth function, 9.6.13
sequence space, 3.3.1 (2) smoothing process, 9.6.18
Sequence Star Principle, 6.4.12 Snow?ake Lemma, 2.3.16
series sum, 5.5.9 (7) space countable at in?nity, 10.9.8
sesquilinear form, 6.1.2 space of bounded elements,
set absorbing another set, 3.4.9 5.5.9 (5)
set in a space, 2.1.4 (3) space of bounded functions,
set lacking a distribution, 5.5.9 (2)
10.10.5 (6) space of bounded operators,
set lacking a functional, 10.8.13 5.1.10 (7)
set lacking a measure, 10.9.4 (5) space of compactly-supported
set of arrival, 1.1.1 distributions, 10.10.5 (9)
set of departure, 1.1.1 space of convergent sequences,
set of second category, 4.7.1 5.5.9 (3)
set supporting a measure, space of distributions of order
at most m, 10.10.8
10.9.4 (5)
set that separates the points space of essentially bounded
of another set, 10.8.9 functions, 5.5.9 (5)
set void of a distribution, space of ?nite-order distributions,
10.10.5 (6) 10.10.8
set void of a functional, 10.8.13 space of functions vanishing
set void of a measure, 10.9.4 (5) at in?nity, 5.5.9 (3)
space of X-valued p-summable
setting in duality, 10.3.3
setting primes, 7.6.5 functions, 5.5.9 (6)
space of p-summable functions,
sheaf, 10.9.11
shift, 10.9.4 (1) 5.5.9 (4)
space of p-summable sequences,
Shilov boundary, Ex. 11.8
Shilov Theorem, 11.2.4 5.5.9 (4)
short sequence, 2.3.5 space of tempered distributions,
?-compact, 10.9.8 10.11.16
signed measure, 10.9.4 (3) space of vanishing sequences,
simple convergence, 9.5.5 (6) 5.5.9 (3)

342

Spectral Decomposition Lemma, strongly holomorphic function,
6.6.6 8.1.5
structure of a subdi?erential,
Spectral Decomposition Theorem,
10.6.3
8.2.12
subadditive functional, 3.4.7 (4)
Spectral Endpoint Theorem, 6.5.5
subcover, 9.6.1
Spectral Mapping Theorem, 8.2.5
subdi?erential, 3.5.1
Spectral Purity Theorem,
sublattice, 10.8.2
11.7.11
sublinear functional, 3.4.6
spectral radius of an operator,
submultiplicative norm, 5.6.1
5.6.6
subnet, 1.3.5 (2)
Spectral Theorem, 11.8.6
subnet in a broad sense, 1.3.5 (2)
spectral value of an element
subrepresentation, 8.2.2
of an algebra, 11.2.1
subspace of a metric space, 4.5.14
spectral value of an operator,
subspace of a topological space,
5.6.13
9.2.17 (1)
spectrum, 10.2.7
subspace of an ordered vector
spectrum of an element
space, 3.2.6 (2)
of an algebra, 11.2.1
subspace topology, 9.2.17 (1)
spectrum of an operator, 5.6.13
Sukhomlinov–Bohnenblust–Sobczyk
spherical layer, 6.2.1
Theorem, 3.7.12
?-algebra, 6.4.13
sum of a family in the sense
?-isomorphism, 11.8.3
of Lp , 5.5.9 (6)
?-linear functional, 2.2.4
sum of germs, 8.1.16
?-representation, 11.8.3
summable family of vectors,
star-shaped set, 3.1.2 (7)
5.5.9 (7)
state, 11.9.1
summable function, 5.5.9 (4)
Steklov condition, 6.3.10 superset, 1.3.3
Steklov Theorem, 6.3.11 sup-norm, 10.8.1
step function, 5.5.9 (6) support function, 10.6.4
Stone Theorem, 10.8.10 support of a distribution,
Stone–Weierstrass Theorem for 10.10.5 (6)
C(Q, C), 11.8.2 support of a functional, 10.8.12
Stone–Weierstrass Theorem for support of a measure, 10.9.4 (5)
C(Q, R), 10.8.17 supporting function, 10.6.4
Strict Separation Theorem, 10.4.8 supremum, 1.2.9
strict subnet, 1.3.5 (2) symmetric Hahn–Banach formula,
strictly positive real, 4.1.3 Ex. 3.10
strong order-unit, 5.5.9 (5) symmetric relation, 1.2.1
strong uniformity, 9.5.5 (6) symmetric set, 3.1.2 (7)
stronger multinorm, 5.3.1 system with integration, 5.5.9 (4)
stronger pretopology, 9.1.2 Szankowski Counterexample,
stronger seminorm, 5.3.3 8.3.13
343

tail ?lter, 1.3.5 (2) topologically complemented
? -dual of a locally convex space, subspace, 7.4.9
topology, 9.1.7
10.2.11
topology compatible with
Taylor Series Expansion Theorem,
duality, 10.4.1
8.1.9
topology compatible with vector
tempered distribution, 10.11.16
structure, 10.1.1
tempered function, 5.1.10 (6),
topology given by open sets,
58; 10.11.6
9.1.12
tempered Radon measure,
topology of a multinormed
10.11.17 (3)
space, 5.2.8
test function, 10.10.1
topology of a uniform space, 9.5.3
test function space, 10.10.1
topology of the distribution
theorem on Hilbert isomorphy,
space, 10.10.6
6.3.17
theorem on the equation AX = B, topology of the test function
space, 10.10.6
2.3.13
theorem on the equation X A = B, total operator, 2.2.1
total set of functionals, 7.4.11 (2)
2.3.8
totally bounded, 4.6.3
theorem on the general form
transitive relation, 1.2.1
of a distribution, 10.10.14
translation, 10.9.4 (1)
theorem on the inverse image
translation of a distribution,
of a vector topology, 10.1.6
10.11.17 (7)
theorem on the repeated Fourier
transpose of an operator, 7.6.2
transform, 10.11.13
trivial topology, 9.1.8 (3)
theorem on the structure
truncator, 9.6.19 (1)
of a locally convex topology,
truncator direction, 10.10.2 (5)
10.2.2
truncator set, 10.10.2
theorem on the structure
twin of a Hilbert space, 6.1.10 (3)
of a vector topology, 10.1.4
twin of a vector space, 2.1.4 (2)
theorem on topologizing
Two Norm Principle, 7.4.16
by a family of mappings,
9.2.16 two-sided ideal, 8.3.3, 132; 11.6.2
Tietze–Urysohn Theorem, Tychono? cube, 9.2.17 (2)
10.8.20 Tychono? product, 9.2.17 (2)
topological isomorphism, 9.2.4 Tychono? space, 9.3.15
topological mapping, 9.2.4 Tychono? Theorem, 9.4.8
Tychono? topology, 9.2.17 (2)
Topological Separation Theorem,
7.5.12 Tychono? uniformity, 9.5.5 (4)
topological space, 9.1.7 T1 -space, 9.3.2
T1 -topology, 9.3.2
topological structure of a convex
set, 7.1.1 T2 -space, 9.3.5
topological subdi?erential, 7.5.8 T3 -space, 9.3.9
topological vector space, 10.1.1 T31 /2 -space, 9.3.15

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