Stereotype space

In functional analysis and related areas of mathematics, stereotype
spaces are topological vector spaces defined by a special variant of
reflexivity condition. They form a class of spaces with a series of
remarkable properties, in particular, this class is very wide (for
instance, it contains all Fréchet spaces and thus, all Banach
spaces), it consists of spaces satisfying a natural condition of
completeness, and it forms a cosmos and a *-autonomous category with
the standard analytical tools for constructing new spaces, like taking
dual spaces, spaces of operators, tensor products, products and
coproducts, limits and colimits, and in addition, immediate subspaces,
and immediate quotient spaces. Mutual embeddings of the main classes
of locally convex spaces. Contents 1 Definition 2 Examples 3 History 4
Pseudocompletion and pseudosaturation 5 Immediate subspaces and
immediate quotient spaces 5.1 Immediate subspaces and envelopes 5.2
Immediate quotient spaces and refinements 6 Category "Ste" of
stereotype spaces 6.1 Kernel and cokernel in the category "Ste" 6.2
"Ste" as a *-autonomous category 6.3 "Ste" as a cosmos 6.4
Grothendieck transformation 6.5 Stereotype approximation property 6.6
Universality of tensor product 7 Applications 8 See also 9 Notes 10
External links 11 References Definition[edit] A stereotype space[1] is
a topological vector space [math]\displaystyle{ X }[/math] over the
field [math]\displaystyle{ \mathbb{C} }[/math] of complex numbers[2]
such that the natural map into the second dual space
[math]\displaystyle{ i:X\to X^{\star\star},\quad i(x)(f)=f(x),\quad
x\in X,\quad f\in X^\star }[/math] is an isomorphism of topological
vector spaces (i.e. a linear and a homeomorphic map). Here the dual
space [math]\displaystyle{ X^\star }[/math] is defined as the space of
all linear continuous functionals [math]\displaystyle{
f:X\to\mathbb{C} }[/math] endowed with the topology of uniform
convergence on totally bounded sets in [math]\displaystyle{ X
}[/math], and the second dual space [math]\displaystyle{
X^{\star\star} }[/math] is the space dual to [math]\displaystyle{
X^{\star} }[/math] in the same sense. The following criterion
holds:[3] a topological vector space [math]\displaystyle{ X }[/math]
is stereotype if and only if it is locally convex and satisfies the
following two conditions: pseudocompleteness: each totally bounded
Cauchy net in [math]\displaystyle{ X }[/math] converges,
pseudosaturateness: each closed convex balanced capacious[4] set
[math]\displaystyle{ D }[/math] in [math]\displaystyle{ X }[/math] is
a neighborhood of zero in [math]\displaystyle{ X }[/math]. The
property of being pseudocomplete is a weakening of the usual notion of
completeness, while the property of being pseudosaturated is a
weakening of the notion of barreledness of a topological vector space.
Examples[edit] The class Ste of stereotype spaces is extremely wide,
so that it will not be a serious exaggeration to say that all
topological vector spaces really used in analysis are stereotype.[5]
Each pseudocomplete barreled space [math]\displaystyle{ X }[/math] (in
particular, each Banach space and each Fréchet space) is
stereotype[6]. Its dual space [math]\displaystyle{ X^\star }[/math]
(which is not barreled, unless [math]\displaystyle{ X }[/math] is a
Montel space) is stereotype as well[7]. There exist stereotype spaces
which are not Mackey spaces[8]. Some simple connections between the
properties of a stereotype space [math]\displaystyle{ X }[/math] and
those of its dual space [math]\displaystyle{ X^\star }[/math] are
expressed in the following list of regularities:[9]
[math]\displaystyle{ X }[/math] is a normed space [math]\displaystyle{
\Longleftrightarrow }[/math] [math]\displaystyle{ X }[/math] is a
Banach space [math]\displaystyle{ \Longleftrightarrow }[/math]
[math]\displaystyle{ X^\star }[/math] is a Smith space;
[math]\displaystyle{ X }[/math] is metrizable [math]\displaystyle{
\Longleftrightarrow }[/math] [math]\displaystyle{ X }[/math] is a
Fréchet space [math]\displaystyle{ \Longleftrightarrow }[/math]
[math]\displaystyle{ X^\star }[/math] is a Brauner space;
[math]\displaystyle{ X }[/math] is barreled [math]\displaystyle{
\Longleftrightarrow }[/math] [math]\displaystyle{ X^\star }[/math] has
the Heine-Borel property; [math]\displaystyle{ X }[/math] is
quasi-barreled [math]\displaystyle{ \Longleftrightarrow }[/math] in
[math]\displaystyle{ X^\star }[/math] if a set [math]\displaystyle{ T
}[/math] is absorbed by each barrel, then [math]\displaystyle{ T
}[/math] is totally bounded; [math]\displaystyle{ X }[/math] is a
Mackey space [math]\displaystyle{ \Longleftrightarrow }[/math] in
[math]\displaystyle{ X^\star }[/math] every [math]\displaystyle{ X
}[/math]-weakly compact set is compact; [math]\displaystyle{ X
}[/math] is a Montel space [math]\displaystyle{ \Longleftrightarrow
}[/math] [math]\displaystyle{ X }[/math] is barreled and has the
Heine-Borel property [math]\displaystyle{ \Longleftrightarrow }[/math]
[math]\displaystyle{ X^\star }[/math] is a Montel space;
[math]\displaystyle{ X }[/math] is a space with a weak topology
[math]\displaystyle{ \Longleftrightarrow }[/math] in
[math]\displaystyle{ X^\star }[/math] every compact set
[math]\displaystyle{ T }[/math] is finite-dimensional;
[math]\displaystyle{ X }[/math] is separable [math]\displaystyle{
\Longleftrightarrow }[/math] in [math]\displaystyle{ X^\star }[/math]
there is a sequence of closed subspaces [math]\displaystyle{ L_n
}[/math] of finite co-dimension with trivial intersection:
[math]\displaystyle{ \bigcap_{n=1}^{\infty}L_n=\{0\} }[/math].
[math]\displaystyle{ X }[/math] has the (classical) approximation
property [math]\displaystyle{ \Longleftrightarrow }[/math]
[math]\displaystyle{ X^\star }[/math] has the (classical)
approximation property; [math]\displaystyle{ X }[/math] is complete
[math]\displaystyle{ \Longleftrightarrow }[/math] [math]\displaystyle{
X^\star }[/math] is co-complete[10] [math]\displaystyle{
\Longleftrightarrow }[/math] [math]\displaystyle{ X^\star }[/math] is
saturated;[11] [math]\displaystyle{ X }[/math] is a Pták space[12]
[math]\displaystyle{ \Longleftrightarrow }[/math] in
[math]\displaystyle{ X^\star }[/math] a subspace [math]\displaystyle{
L }[/math] is closed if it has the closed intersection
[math]\displaystyle{ L\cap K }[/math] with each compact set
[math]\displaystyle{ K\subseteq X^\star }[/math]; [math]\displaystyle{
X }[/math] is hypercomplete[13] [math]\displaystyle{
\Longleftrightarrow }[/math] in [math]\displaystyle{ X^\star }[/math]
an absolutely convex set [math]\displaystyle{ B }[/math] is closed if
it has the closed intersection [math]\displaystyle{ B\cap K }[/math]
with each compact set [math]\displaystyle{ K\subseteq X^\star
}[/math]. Counterexamples: 1. If a metrizable locally convex space
[math]\displaystyle{ X }[/math] is not complete, then it is not
stereotype.[14] 2. If [math]\displaystyle{ X }[/math] is an infinite
dimensional Banach space, and [math]\displaystyle{ Y=X'_\sigma
}[/math] is its dual space (of linear continuous functionals
[math]\displaystyle{ f:X\to{\mathbb C} }[/math]) considered with the
[math]\displaystyle{ X }[/math]-weak topology, then
[math]\displaystyle{ Y }[/math] is not stereotype.[15] History[edit]
The first results on this type of reflexivity of topological vector
spaces were obtained by M. F. Smith[16][17] in 1952. Further
investigations were conducted by B. S. Brudovskii, [18] W. C.
Waterhouse,[19] K. Brauner,[20] S. S. Akbarov,[21][22][23][24] and E.
T. Shavgulidze.[25] The term "stereotype space" was introduced by S.
S. Akbarov in 1995[26]. The main properties of the category of
stereotype spaces were described by S. S. Akbarov in his series of
works of 1995-2017. Pseudocompletion and pseudosaturation[edit] Each
locally convex space [math]\displaystyle{ X }[/math] can be
transformed into a stereotype space with the help of two standard
operations, pseudocompletion and pseudosaturation, defined by the
following two propositions. Theorem.[27] For each locally convex space
[math]\displaystyle{ X }[/math] there exists a pseudocomplete locally
convex space [math]\displaystyle{ X^\triangledown }[/math] and a
linear continuous mapping [math]\displaystyle{ {\triangledown}_X :X
\to X^{\triangledown} }[/math] such that for every pseudocomplete
locally convex space [math]\displaystyle{ Y }[/math] and for every
linear continuous mapping [math]\displaystyle{ \varphi:X\to Y }[/math]
there is a unique linear continuous mapping [math]\displaystyle{
\varphi^\triangledown:X^\triangledown\to Y }[/math] such that
[math]\displaystyle{
\varphi^\triangledown\circ{\triangledown}_X=\varphi }[/math]. The
space [math]\displaystyle{ X^\triangledown }[/math] is called a
pseudocompletion of the space [math]\displaystyle{ X }[/math]. It is
unique up to an isomorphism of locally convex spaces. For each linear
continuous mapping of locally convex spaces [math]\displaystyle{
\varphi :X\to Y }[/math] there is a unique linear continuous mapping
[math]\displaystyle{ \varphi^{\triangledown} :X^{\triangledown} \to
Y^{\triangledown} }[/math] such that [math]\displaystyle{
\triangledown_Y\circ\varphi=\varphi^\triangledown\circ\triangledown_X
}[/math], and the correspondence [math]\displaystyle{
\varphi\mapsto\varphi^{\triangledown} }[/math] can be defined as a
(covariant) functor. The pseudocompletion [math]\displaystyle{
X^{\triangledown} }[/math] can be defined as an envelope of the
locally convex space [math]\displaystyle{ X }[/math] in the class
[math]\displaystyle{ \operatorname{PC} }[/math] of all pseudocomplete
locally convex spaces with respect to the same class
[math]\displaystyle{ \operatorname{PC} }[/math]:[28]
[math]\displaystyle{
X^{\triangledown}=\operatorname{Env}_{\operatorname{PC}}^{\operatorname{PC}}
X }[/math] One can imagine the pseudocompletion of
[math]\displaystyle{ X }[/math] as the "nearest to
[math]\displaystyle{ X }[/math] from the outside" pseudocomplete
locally convex space, so that the operation [math]\displaystyle{
X\mapsto X^\triangledown }[/math] adds to [math]\displaystyle{ X
}[/math] some supplementary elements, but does not change the topology
of [math]\displaystyle{ X }[/math] (like the usual operation of
completion). Theorem.[29] For each locally convex space
[math]\displaystyle{ X }[/math] there is a pseudosaturated locally
convex space [math]\displaystyle{ X^\vartriangle }[/math] and a linear
continuous mapping [math]\displaystyle{ {\vartriangle}_X
:X^{\vartriangle} \to X }[/math] such that for each pseudosaturated
locally convex space [math]\displaystyle{ Y }[/math] and for each
linear continuous mapping [math]\displaystyle{ \varphi:Y\to X }[/math]
there is a unique linear continuous mapping [math]\displaystyle{
\varphi^\vartriangle:Y\to X^\vartriangle }[/math] such that
[math]\displaystyle{
{\vartriangle}_X\circ\varphi^\vartriangle=\varphi. }[/math] The space
[math]\displaystyle{ X^\vartriangle }[/math] is called a
pseudosaturation of the space [math]\displaystyle{ X }[/math]. It is
unique up to an isomorphism of locally convex spaces. For each linear
continuous mapping of locally convex spaces [math]\displaystyle{
\varphi :Y\to X }[/math] there is a unique linear continuous mapping
[math]\displaystyle{ \varphi^{\vartriangle} :Y^{\vartriangle} \to
X^{\vartriangle} }[/math] such that [math]\displaystyle{ \varphi\
\circ\vartriangle_Y=\vartriangle_X\circ\ \varphi^\vartriangle
}[/math], and the correspondence [math]\displaystyle{
\varphi\mapsto\varphi^{\vartriangle} }[/math] can be defined as a
(covariant) functor. The pseudosaturation [math]\displaystyle{
X^{\vartriangle} }[/math] can be defined as a refinement of the
locally convex space [math]\displaystyle{ X }[/math] in the class
[math]\displaystyle{ \operatorname{PS} }[/math] of all pseudosaturated
locally convex spaces with respect to the same class
[math]\displaystyle{ \operatorname{PS} }[/math]:[30]
[math]\displaystyle{
X^{\vartriangle}=\operatorname{Ref}_{\operatorname{PS}}^{\operatorname{PS}}
X. }[/math] One can imagine the pseudosaturation of
[math]\displaystyle{ X }[/math] as the "nearest to
[math]\displaystyle{ X }[/math] from the inside" pseudosaturated
locally convex space, so that the operation [math]\displaystyle{
X\mapsto X^\vartriangle }[/math] strengthens the topology of
[math]\displaystyle{ X }[/math], but does not change the elements of
[math]\displaystyle{ X }[/math]. If [math]\displaystyle{ X }[/math] is
a pseudocomplete locally convex space, then its pseudosaturation
[math]\displaystyle{ X^\vartriangle }[/math] is stereotype. Dually, if
[math]\displaystyle{ X }[/math] is a pseudosaturated locally convex
space, then its pseudocompletion [math]\displaystyle{ X^\triangledown
}[/math] is stereotype. For arbitrary locally convex space
[math]\displaystyle{ X }[/math] the spaces [math]\displaystyle{
X^{\vartriangle\triangledown} }[/math] and [math]\displaystyle{
X^{\triangledown\vartriangle} }[/math] are stereotype.[31] Immediate
subspaces and immediate quotient spaces[edit] The idea of subspace
(and of quotient space) in stereotype theory leads to more complicated
results than in the theory of locally convex spaces. Immediate
subspaces and envelopes[edit] The notion of immediate subspace gives a
"concrete description" of the abstract notion of immediate
monomorphism[32], or, what is equivalent in this situation[33], strong
monomorphism[34] in the category Ste. Surprisingly, this description
does not coincide with the construction of closed subspace in the
category LocConv of locally convex spaces. Suppose
[math]\displaystyle{ Y }[/math] is a subset in a stereotype space
[math]\displaystyle{ X }[/math] endowed with a structure of a
stereotype space in such a way that the set-theoretic inclusion
[math]\displaystyle{ Y\subseteq X }[/math] is a morphism of stereotype
spaces (i.e. a continuous linear map). Then [math]\displaystyle{ Y
}[/math] is called a subspace of the stereotype space
[math]\displaystyle{ X }[/math], with the notation
[math]\displaystyle{ Y\hookrightarrow X }[/math]. Suppose we have a
chain of stereotype subspaces [math]\displaystyle{ Z\hookrightarrow
Y\hookrightarrow X }[/math], and the first mapping
[math]\displaystyle{ Z\hookrightarrow Y }[/math] is a bimorphism of
stereotype spaces. Then the space [math]\displaystyle{ Y }[/math] is
called a mediator of the subspace [math]\displaystyle{ Z }[/math] in
the space [math]\displaystyle{ X }[/math]. A subspace
[math]\displaystyle{ Z }[/math] in a stereotype space
[math]\displaystyle{ X }[/math] is called an immediate subspace in
[math]\displaystyle{ X }[/math], with the notation
[math]\displaystyle{ Z\overset{\circ}{\hookrightarrow} X }[/math], if
it has no non-trivial mediators, i.e. for any mediator
[math]\displaystyle{ Y }[/math] of [math]\displaystyle{ Z }[/math] in
[math]\displaystyle{ X }[/math] the inclusion [math]\displaystyle{
Z\hookrightarrow Y }[/math] is an isomorphism. Examples: 1. An
immediate subspace [math]\displaystyle{ Z }[/math] in a stereotype
space [math]\displaystyle{ X }[/math] is said to be closed, if
[math]\displaystyle{ Z }[/math] (as a set) is closed in
[math]\displaystyle{ X }[/math] (as a topological space). If
[math]\displaystyle{ Y }[/math] is a closed subspace in a stereotype
space [math]\displaystyle{ X }[/math] (as in a locally convex space),
then its pseudosaturation [math]\displaystyle{ Z=Y^\vartriangle
}[/math] is a closed immediate subspace in [math]\displaystyle{ X
}[/math]. All closed immediate subspaces have this form. 2. There are
stereotype spaces [math]\displaystyle{ X }[/math] with closed
immediate subspaces [math]\displaystyle{ Z=Y^\vartriangle }[/math]
whose topology is not inherited from [math]\displaystyle{ X
}[/math][35] (this is one of the qualitative differences with the
category LocConv of locally convex spaces).[36] 3. In contrast to the
category LocConv of locally convex spaces in the category Ste the
immediate subspaces are not always closed.[36] Theorem.[37] For any
set [math]\displaystyle{ M }[/math] in a stereotype space
[math]\displaystyle{ X }[/math] there is a minimal immediate subspace
[math]\displaystyle{ \operatorname{Env}^XM }[/math] in
[math]\displaystyle{ X }[/math], containing [math]\displaystyle{ M
}[/math]: (i) [math]\displaystyle{
M\subseteq\operatorname{Env}^XM\overset{\circ}{\hookrightarrow} X
}[/math] (ii) [math]\displaystyle{ \forall Z \quad (M\subseteq
Z\overset{\circ}{\hookrightarrow} X\ \Rightarrow\
\operatorname{Env}^XM\subseteq Z) }[/math], and this subspace
[math]\displaystyle{ \operatorname{Env}^XM }[/math] is an immediate
subspace in each immediate subspace, containing [math]\displaystyle{ M
}[/math]: (iii) [math]\displaystyle{ \forall Z \quad (M\subseteq
Z\overset{\circ}{\hookrightarrow} X\ \Rightarrow\
\operatorname{Env}^XM\overset{\circ}{\hookrightarrow} Z) }[/math], The
subspace [math]\displaystyle{ \operatorname{Env}^XM }[/math] is called
an envelope of the set [math]\displaystyle{ M }[/math] in the
stereotype space [math]\displaystyle{ X }[/math]. Theorem.[38] Each
set [math]\displaystyle{ M }[/math] in a stereotype space
[math]\displaystyle{ X }[/math] is a total set[39] in its envelope
[math]\displaystyle{ \operatorname{Env}^XM }[/math]. If
[math]\displaystyle{ {\mathbb C}_M }[/math] denotes the space of all
functions [math]\displaystyle{ \alpha:M\to{\mathbb C} }[/math] with
finite support, endowed with the strongest locally convex topology,
and the mapping [math]\displaystyle{ \varphi:{\mathbb C}_M\to X
}[/math] acts by the formula [math]\displaystyle{
\varphi(\alpha)=\sum_{x\in M}\alpha(x)\cdot x }[/math], then the
envelope [math]\displaystyle{ \operatorname{Env}^XM }[/math] coincides
with the abstract categorical envelope of the space
[math]\displaystyle{ {\mathbb C}_M }[/math] in the class
[math]\displaystyle{ \operatorname{Epi} }[/math] of all epimorphisms
in the category Ste with respect to the morphism [math]\displaystyle{
\varphi }[/math]:[40] [math]\displaystyle{
\operatorname{Env}^XM=\operatorname{Env}_\varphi^{\operatorname{Epi}}{\mathbb
C}_M }[/math] Immediate quotient spaces and refinements[edit] Dually,
the notion of immediate quotient space gives a "concrete description"
of the abstract notion of immediate epimorphism[41], or, what is
equivalent here[33], strong epimorphism[42] in the category Ste. Like
in the situation with monomorphisms, this description does not
coincide with the construction of quotient space in the category
LocConv of locally convex spaces. Let [math]\displaystyle{ E }[/math]
be a closed subspace (in the usual sense) in a stereotype space
[math]\displaystyle{ X }[/math]. Consider a topology
[math]\displaystyle{ \tau }[/math] on the quotient space
[math]\displaystyle{ X/E }[/math], which is majorized by the usual
quotient topology of [math]\displaystyle{ X/E }[/math]. Let
[math]\displaystyle{ (X/E)^\blacktriangledown }[/math] be a completion
of [math]\displaystyle{ X/E }[/math] with respect to the topology
[math]\displaystyle{ \tau }[/math]. Suppose [math]\displaystyle{ Y
}[/math] is a subset in the locally convex space [math]\displaystyle{
(X/E)^\blacktriangledown }[/math] which contains [math]\displaystyle{
X/E }[/math] and at the same time is a stereotype space. Then
[math]\displaystyle{ Y }[/math] is called a quotient space of the
stereotype space [math]\displaystyle{ X }[/math], with the notation
[math]\displaystyle{ X\rightarrowtail Y }[/math]. Suppose we have two
quotient spaces [math]\displaystyle{ X\rightarrowtail Y }[/math] and
[math]\displaystyle{ X\rightarrowtail Z }[/math]. It is said that the
[math]\displaystyle{ Y }[/math] subordinates [math]\displaystyle{ Z
}[/math] (notation: [math]\displaystyle{ Y\ge Z }[/math]) if there is
a morphism [math]\displaystyle{ \varkappa:Y\to Z }[/math] such that
[math]\displaystyle{ \upsilon_Z=\varkappa\circ\upsilon_Y }[/math]
(where [math]\displaystyle{ \upsilon_Z:X\to Z }[/math] and
[math]\displaystyle{ \upsilon_Y:X\to Y }[/math] are the natural
mappings). Suppose that the quotient space [math]\displaystyle{
X\rightarrowtail Y }[/math] subordinates the quotient space
[math]\displaystyle{ X\rightarrowtail Z }[/math] (i.e.
[math]\displaystyle{ Y\ge Z }[/math]) and the corresponding morphism
[math]\displaystyle{ \varkappa:Y\to Z }[/math] is a bimorphism. Then
the quotient space [math]\displaystyle{ Y }[/math] is called a
mediator of the quotient space [math]\displaystyle{ Z }[/math] of the
space [math]\displaystyle{ X }[/math]. A quotient space
[math]\displaystyle{ Z }[/math] of a stereotype space
[math]\displaystyle{ X }[/math] is called an immediate quotient space
of [math]\displaystyle{ X }[/math], with the notation
[math]\displaystyle{ X\overset{\circ}{\rightarrowtail} Z }[/math], if
it has no non-trivial mediators, i.e. for any mediator
[math]\displaystyle{ Y }[/math] of [math]\displaystyle{ Z }[/math] the
morphism [math]\displaystyle{ \varkappa:Y\to Z }[/math] is an
isomorphism. Examples: 1. An immediate quotient space
[math]\displaystyle{ Z }[/math] of a stereotype space
[math]\displaystyle{ X }[/math] is said to be open, if the
corresponding map [math]\displaystyle{ X\to Z }[/math] is open[43]. If
[math]\displaystyle{ E }[/math] is a closed subspace in a stereotype
space [math]\displaystyle{ X }[/math], then the pseudocompletion
[math]\displaystyle{ Z=(X/E)^\triangledown }[/math] of the (locally
convex) quotient space [math]\displaystyle{ X/E }[/math] is an open
immediate quotient space of [math]\displaystyle{ X }[/math]. All open
immediate quotient spaces have this form. 2. There are stereotype
spaces [math]\displaystyle{ X }[/math] with immediate quotient spaces
[math]\displaystyle{ Z }[/math] which cannot be represented in the
form [math]\displaystyle{ Z=X/E }[/math].[44] 3. In contrast to the
category LocConv of locally convex spaces in the category Ste
immediate quotient spaces are not always open.[44] Theorem.[45] For
any set [math]\displaystyle{ F }[/math] of linear continuous
functionals on a stereotype space [math]\displaystyle{ X }[/math]
there is a minimal immediate quotient space [math]\displaystyle{
\operatorname{Ref}^XF }[/math] of [math]\displaystyle{ X }[/math] to
which all functionals [math]\displaystyle{ F }[/math] can be extended:
(i) [math]\displaystyle{
X\overset{\circ}{\rightarrowtail}\operatorname{Ref}^XF\ \&\
(\operatorname{Ref}^XF)^\star\supseteq F }[/math] (ii)
[math]\displaystyle{ \forall Z \quad
\big((X\overset{\circ}{\rightarrowtail}Z\ \&\ Z^\star\supseteq F)\
\Rightarrow\ Z\ge\operatorname{Ref}^XF\big) }[/math], and this
quotient space [math]\displaystyle{ \operatorname{Ref}^XF }[/math] is
(up to an isomorphism) an immediate quotient space of each immediate
quotient space, to which the functionals [math]\displaystyle{ F
}[/math] are extended: (iii) [math]\displaystyle{ \forall Z \quad
\big((X\overset{\circ}{\rightarrowtail}Z\ \&\ Z^\star\supseteq F)\
\Rightarrow\
Z\overset{\circ}{\rightarrowtail}\operatorname{Ref}^XF\big) }[/math].
The quotient space [math]\displaystyle{ \operatorname{Ref}^XF }[/math]
is called a refinement of the set [math]\displaystyle{ F }[/math] on
the stereotype space [math]\displaystyle{ X }[/math]. Theorem.[46]
Each set [math]\displaystyle{ F }[/math] of linear continuous
functionals on a stereotype space [math]\displaystyle{ X }[/math] is a
total set[47] on its refinement [math]\displaystyle{
\operatorname{Ref}^XF }[/math]. If [math]\displaystyle{ {\mathbb C}^F
}[/math] denotes the space of all functions [math]\displaystyle{
f:M\to{\mathbb C} }[/math], endowed with the topology of pointwise
convergence, and the mapping [math]\displaystyle{ \varphi:X\to{\mathbb
C}^F }[/math] acts by the formula [math]\displaystyle{
\varphi(x)(f)=f(x) }[/math], then the refinement [math]\displaystyle{
\operatorname{Ref}^XF }[/math] coincides with the abstract categorical
refinement of the space [math]\displaystyle{ {\mathbb C}^F }[/math] in
the class [math]\displaystyle{ \operatorname{Mono} }[/math] of all
monomorphisms in the category Ste by means of the morphism
[math]\displaystyle{ \varphi }[/math]:[40] [math]\displaystyle{
\operatorname{Ref}^XF=\operatorname{Ref}_\varphi^{\operatorname{Mono}}{\mathbb
C}^F }[/math] Category "Ste" of stereotype spaces[edit] The class Ste
of stereotype spaces forms a category with linear continuous maps as
morphisms and possesses the following properties: Ste is
pre-abelian[48]; Ste is bicomplete[49]; Ste is autodual with respect
to the functor [math]\displaystyle{ X\mapsto X^\star }[/math][50]; Ste
is a category with nodal decomposition[51]. Ste is a *-autonomous
category[52]; Ste is a cosmos. Kernel and cokernel in the category
"Ste"[edit] Ste is a pre-abelian category: each morphism
[math]\displaystyle{ \varphi:X\to Y }[/math] in the category Ste has a
kernel [math]\displaystyle{ \operatorname{Ker}\varphi=\{x\in
X:\varphi(x)=0\}^\vartriangle=\operatorname{Env}^X\{x\in
X:\varphi(x)=0\} }[/math] and a cokernel [math]\displaystyle{
\operatorname{Coker}\varphi=\Big(Y\Big/\overline{\varphi(X)}\Big)^\triangledown=\operatorname{Ref}^Y\{f\in
Y^\star:\ f\circ\varphi=0\}. }[/math] As a corollary,
[math]\displaystyle{ \varphi }[/math] has an image
[math]\displaystyle{ \operatorname{Im}\varphi }[/math] and a coimage
[math]\displaystyle{ \operatorname{Coim}\varphi }[/math] as well. The
following natural identities hold:[48] [math]\displaystyle{
\begin{align} & (\operatorname{Ker}\varphi)^\star
=\operatorname{Coker}(\varphi^\star), & &
(\operatorname{Coker}\varphi)^\star
=\operatorname{Ker}(\varphi^\star), \\[5pt] &
(\operatorname{Im}\varphi)^\star =\operatorname{Coim}(\varphi^\star),
& & (\operatorname{Coim}\varphi)^\star
=\operatorname{Im}(\varphi^\star), \\[5pt] &
(\operatorname{Ker}\varphi)^{\perp} =\operatorname{Im}(\varphi^\star),
& & (\operatorname{Im}\varphi)^{\perp}
=\operatorname{Ker}(\varphi^\star), \\[5pt] &
\operatorname{Ker}\varphi =(\operatorname{Im}(\varphi^\star))^{\perp},
& & \operatorname{Im}\varphi
=(\operatorname{Ker}(\varphi^\star))^{\perp}, \end{align} }[/math]
where [math]\displaystyle{ Q^{\perp} }[/math] denotes the
pseudosaturation of the annihilator of the subspace
[math]\displaystyle{ Q\subseteq P }[/math] in the dual space
[math]\displaystyle{ P^\star }[/math]: [math]\displaystyle{
Q^\perp=\{f\in P^\star: \
f|_Q=0\}^\vartriangle=\operatorname{Env}^{P^\star}\{f\in P^\star: \
f|_Q=0\} }[/math]. "Ste" as a *-autonomous category[edit] For any two
stereotype spaces [math]\displaystyle{ X }[/math] and
[math]\displaystyle{ Y }[/math] the stereotype space of operators
[math]\displaystyle{ Y\oslash X }[/math] from [math]\displaystyle{ X
}[/math] into [math]\displaystyle{ Y }[/math], is defined as the
pseudosaturation of the space [math]\displaystyle{
\operatorname{L}(X,Y) }[/math] of all linear continuous maps
[math]\displaystyle{ \varphi:X\to Y }[/math] endowed with the topology
of uniform convergeance on totally bounded sets. The space
[math]\displaystyle{ Y\oslash X }[/math] is stereotype. It defines two
natural tensor products [math]\displaystyle{ X\circledast Y:=
(X^\star\oslash Y)^\star, }[/math] [math]\displaystyle{ X\odot Y :=
Y\oslash X^\star. }[/math] Theorem. In the category Ste the following
natural identities hold:[52][53][54]: [math]\displaystyle{
\mathbb{C}\circledast X\cong X\cong X\circledast \mathbb{C}, }[/math]
[math]\displaystyle{ \mathbb{C}\odot X\cong X\cong X\odot\mathbb{C},
}[/math] [math]\displaystyle{ X\circledast Y\cong Y\circledast X,
}[/math] [math]\displaystyle{ X\odot Y\cong Y\odot X, }[/math]
[math]\displaystyle{ (X\circledast Y)\circledast Z\cong X\circledast
(Y\circledast Z), }[/math] [math]\displaystyle{ (X\odot Y)\odot Z\cong
X\odot (Y\odot Z), }[/math] [math]\displaystyle{ (X\circledast
Y)^\star\cong Y^\star\odot X^\star, }[/math] [math]\displaystyle{
(X\odot Y)^\star\cong Y^\star\circledast X^\star, }[/math]
[math]\displaystyle{ X\oslash Y\cong Y^\star\oslash X^\star, }[/math]
[math]\displaystyle{ X\oslash (Y\circledast Z)\cong (X\oslash
Y)\oslash Z, }[/math] [math]\displaystyle{ (X\odot Y)\oslash Z\cong
X\odot(Y\oslash Z). }[/math] In particular, Ste is a symmetric
monoidal category with respect to the bifunctor [math]\displaystyle{
\odot }[/math], a closed symmetric monoidal category with respect to
the bifunctor [math]\displaystyle{ \circledast }[/math] and the
internal hom-functor [math]\displaystyle{ \oslash }[/math], and a
*-autonomous category: [math]\displaystyle{ X^{\star\star}\cong X,
}[/math] [math]\displaystyle{ X^\star\oslash (Y\circledast Z)\cong
(X\circledast Y)^\star\oslash Z. }[/math] Examples: 1. If
[math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math]
are Fréchet spaces, then their stereotype tensor product
[math]\displaystyle{ X\circledast Y }[/math] coincides with the usual
projective tensor product [math]\displaystyle{ X\hat{\otimes}Y
}[/math] of locally convex spaces [math]\displaystyle{ X }[/math] and
[math]\displaystyle{ Y }[/math].[55] 2. If [math]\displaystyle{ X
}[/math] and [math]\displaystyle{ Y }[/math] are Fréchet spaces and
at least one of them possesses the (classical) approximation property,
then their stereotype tensor product [math]\displaystyle{ X\odot Y
}[/math] coincides with the usual injective tensor product
[math]\displaystyle{ X\check{\otimes}Y }[/math] of locally convex
spaces [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y
}[/math].[56] "Ste" as a cosmos[edit] Ste is a bicomplete category:
each small diagram [math]\displaystyle{ F:J\to\, }[/math]Ste has a
colimit (or direct limit), [math]\displaystyle{ \lim_{j\to\infty}F_j
}[/math], which coincides with the pseudocompletion of the
corresponding colimit in the category LocConv of locally convex
spaces[57] [math]\displaystyle{
\lim_{j\to\infty}F_j=\Big(\overset{\operatorname{\bf
LocConv}}{\lim_{j\to\infty}}F_j\Big)^\triangledown }[/math], and a
limit (or inverse limit), [math]\displaystyle{ \lim_{\infty\gets j}F_j
}[/math], which coincides with the pseudosaturation of the
corresponding limit in LocConv[57] [math]\displaystyle{
\lim_{\infty\gets j}F_j=\Big(\overset{\operatorname{\bf
LocConv}}{\lim_{\infty\gets j}}F_j\Big)^\vartriangle }[/math].
However, the direct sum and the direct product in Ste coincide with
the corresponding constructions in LocConv: [math]\displaystyle{
\bigoplus_{i\in I} X_i=\overset{\operatorname{\bf
LocConv}}{\bigoplus_{i\in I}} X_i,\quad \prod_{i\in I}
X_i=\overset{\operatorname{\bf LocConv}}{\prod_{i\in I}X_i}. }[/math]
Together with the symmetric closed monoidal structure, the existence
of limits and colimits implies the following property: Theorem. The
category Ste is a cosmos. The following natural identities
hold:[52][54] [math]\displaystyle{ \Big(\bigoplus_{i\in I}
X_i\Big)^\star\cong \prod_{i\in I} X_i^\star }[/math]
[math]\displaystyle{ \Big(\prod_{i\in I} X_i\Big)^\star\cong
\bigoplus_{i\in I} X_i^\star }[/math] [math]\displaystyle{ Y\oslash
\Big(\bigoplus_{i\in I} X_i\Big)\cong \prod_{i\in I} ( Y\oslash X_i)
}[/math] [math]\displaystyle{ \Big(\prod_{j\in J} Y_j\Big)\oslash
X\cong \prod_{j\in J} ( Y_j\oslash X) }[/math] [math]\displaystyle{
\Big(\bigoplus_{i\in I} X_i\Big)\circledast \Big(\bigoplus_{j\in J}
Y_j\Big) \cong \bigoplus_{i\in I,j\in J} ( X_i\circledast Y_j)
}[/math] [math]\displaystyle{ \Big(\prod_{i\in I} X_i\Big)\odot
\Big(\prod_{j\in J} Y_j\Big) \cong \prod_{i\in I,j\in J} ( X_i\odot
Y_j) }[/math] [math]\displaystyle{ \Big(\lim_{i\to\infty}
X_i\Big)^\star\cong \lim_{\infty\gets i} X_i^\star }[/math]
[math]\displaystyle{ \Big(\lim_{\infty\gets i} X_i\Big)^\star\cong
\lim_{i\to\infty} X_i^\star }[/math] [math]\displaystyle{ Y\oslash
\Big(\lim_{i\to\infty} X_i\Big)\cong \lim_{\infty\gets i} (Y\oslash
X_i) }[/math] [math]\displaystyle{ \Big(\lim_{\infty\gets j}
Y_j\Big)\oslash X\cong \lim_{\infty\gets j} (Y_j\oslash X) }[/math]
[math]\displaystyle{ \Big(\lim_{i\to\infty} X_i\Big)\circledast
\Big(\lim_{j\to\infty} Y_j\Big) \cong \lim_{i,j\to\infty} (
X_i\circledast Y_j) }[/math] [math]\displaystyle{
\Big(\lim_{\infty\gets i} X_i\Big)\odot \Big(\lim_{\infty\gets j}
Y_j\Big) \cong \lim_{\infty\gets i,j} ( X_i\odot Y_j) }[/math]
Grothendieck transformation[edit] If [math]\displaystyle{ X }[/math]
and [math]\displaystyle{ Y }[/math] are stereotype spaces then for
each elements [math]\displaystyle{ x\in X }[/math] and
[math]\displaystyle{ y\in Y }[/math] the formula [math]\displaystyle{
(x\circledast y)(\varphi)=\varphi(y)(x),\qquad \varphi\in
X^\star\oslash Y }[/math] defines an elementary tensor
[math]\displaystyle{ x\circledast y\in X\circledast Y=(X^\star\oslash
Y)^\star }[/math], and the formula [math]\displaystyle{ (x\odot
y)(f)=f(x)\cdot y,\qquad f\in X^\star }[/math] defines an elementary
tensor [math]\displaystyle{ x\odot y\in X\odot Y=Y\oslash X^\star
}[/math] Theorem.[58] For each stereotype spaces [math]\displaystyle{
X }[/math] and [math]\displaystyle{ Y }[/math] there is a unique
linear continuous map [math]\displaystyle{ \Gamma_{X,Y}: X\circledast
Y\to X\odot Y }[/math] which turns elementary tensors
[math]\displaystyle{ x\circledast y }[/math] into elementary tensors
[math]\displaystyle{ x\odot y }[/math]: [math]\displaystyle{
\Gamma_{X,Y}(x\circledast y)=x\odot y,\qquad x\in X, \ y\in Y.
}[/math] The family of maps [math]\displaystyle{ \Gamma_{X,Y}:
X\circledast Y\to X\odot Y }[/math] defines a natural transformation
of the bifunctor [math]\displaystyle{ \circledast }[/math] into the
bifunctor [math]\displaystyle{ \odot }[/math]. The map
[math]\displaystyle{ \Gamma_{X,Y} }[/math] is called the Grothendieck
transformation. Stereotype approximation property[edit] A stereotype
space [math]\displaystyle{ X }[/math] is said to have the stereotype
approximation property, if each linear continuous map
[math]\displaystyle{ \varphi:X\to X }[/math] can be approximated in
the stereotype space of operators [math]\displaystyle{ X\oslash X
}[/math] by the linear continuous maps of finite rank. This condition
is weaker than the existence of the Schauder basis, but formally
stronger than the classical approximation property (however, it is not
clear (2017) whether the stereotype approximation property coincides
with the classical one, or not). Theorem.[59] For a stereotype space
[math]\displaystyle{ X }[/math] the following conditions are
equivalent: (i) [math]\displaystyle{ X }[/math] has the stereotype
approximation property; (ii) the Grothendieck transformation
[math]\displaystyle{ \Gamma_{X,X^\star}:X\circledast X^\star\to X\odot
X^\star }[/math] is a monomorphism (in the category Ste); (iii) the
Grothendieck transformation [math]\displaystyle{
\Gamma_{X,X^\star}:X\circledast X^\star\to X\odot X^\star }[/math] is
an epimorphism (in the category Ste); (iv) for any stereotype space
[math]\displaystyle{ Y }[/math] the Grothendieck transformation
[math]\displaystyle{ \Gamma_{X,Y}:X\circledast Y\to X\odot Y }[/math]
is a monomorphism (in the category Ste); (v) for any stereotype space
[math]\displaystyle{ Y }[/math] the Grothendieck transformation
[math]\displaystyle{ \Gamma_{X,Y}:X\circledast Y\to X\odot Y }[/math]
is an epimorphism (in the category Ste). Theorem.[60] If two
stereotype spaces [math]\displaystyle{ X }[/math] and
[math]\displaystyle{ Y }[/math] have the stereotype approximation
property, then the spaces [math]\displaystyle{ X\oslash Y }[/math],
[math]\displaystyle{ X\circledast Y }[/math] and [math]\displaystyle{
X\odot Y }[/math] have the stereotype approximation property as well.
In particular, if [math]\displaystyle{ X }[/math] has the stereotype
approximation property, then the same is true for [math]\displaystyle{
X^\star }[/math] and for [math]\displaystyle{ X\oslash X }[/math].
Universality of tensor product[edit] For any stereotype spaces
[math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math],
[math]\displaystyle{ Z }[/math] a bilinear map [math]\displaystyle{
\beta:X\times Y\to Z }[/math] is said to be continuous (as a bilinear
map of stereotype spaces) if 1) for each neighborhood of zero
[math]\displaystyle{ W\subseteq Z }[/math] and for each compact set
[math]\displaystyle{ S\subseteq X }[/math] there exists a neighborhood
of zero [math]\displaystyle{ V\subseteq Y }[/math] such that
[math]\displaystyle{ \beta(S,V)\subseteq W }[/math], and 2) for each
neighborhood of zero [math]\displaystyle{ W\subseteq Z }[/math] and
for each compact set [math]\displaystyle{ T\subseteq Y }[/math] there
exists a neighborhood of zero [math]\displaystyle{ U\subseteq X
}[/math] such that [math]\displaystyle{ \beta(U,T)\subseteq W
}[/math]. Examples: 1. For any stereotype space [math]\displaystyle{ X
}[/math] the pairing [math]\displaystyle{ (x,f)\in X\times X^\star\to
f(x)\in {\mathbb C} }[/math] is a continuous bilinear map. 2. For any
two stereotype spaces [math]\displaystyle{ X }[/math] and
[math]\displaystyle{ Y }[/math] the map [math]\displaystyle{ (x,y)\in
X\times Y\to x\circledast y\in X\circledast Y }[/math] is a continuous
bilinear map. 3. For any two stereotype spaces [math]\displaystyle{ X
}[/math] and [math]\displaystyle{ Y }[/math] the map
[math]\displaystyle{ (x,y)\in X\times Y\to x\odot y\in X\odot Y
}[/math] is a continuous bilinear map. Theorem.[61] For any stereotype
spaces [math]\displaystyle{ X }[/math], [math]\displaystyle{ Y
}[/math], [math]\displaystyle{ Z }[/math] and for any continuous
bilinear map [math]\displaystyle{ \beta:X\times Y\to Z }[/math] there
exists a unique continuous linear map [math]\displaystyle{
\widetilde{\beta}:X\circledast Y\to Z }[/math] such that
[math]\displaystyle{ \widetilde{\beta}\circ\iota=\beta }[/math], where
[math]\displaystyle{ \iota(x,y)=x\circledast y, \ x\in X, \ y\in Y
}[/math]. Corollary.[60] For any stereotype space [math]\displaystyle{
X }[/math] the pairing [math]\displaystyle{ (x,f)\in X\times
X^\star\to f(x)\in {\mathbb C} }[/math] has a unique extension to a
linear continuous functional [math]\displaystyle{
\operatorname{cont}:X\circledast X^\star\to{\mathbb C} }[/math]. This
functional in its turn can be represented as a trace of the operators
[math]\displaystyle{ \varphi:X\to X }[/math] occurring as images of
the tensors [math]\displaystyle{ \alpha\in X\circledast X^\star
}[/math] under the Grothendieck transformation [math]\displaystyle{
\Gamma_{X,X^\star}: X\circledast X^\star\to X\odot X^\star=X\oslash X
}[/math] if and only if the space [math]\displaystyle{ X }[/math] has
the stereotype approximation property. Applications[edit] Being a
symmetric monoidal category, Ste generates the notions of a stereotype
algebra (as a monoid in Ste) and a stereotype module (as a module in
Ste over such a monoid), and it turns out that for each stereotype
algebra [math]\displaystyle{ A }[/math] the categories
[math]\displaystyle{ A }[/math]Ste and Ste[math]\displaystyle{ A
}[/math] of left and right stereotype modules over
[math]\displaystyle{ A }[/math] have the structure of enriched
categories over Ste.[62] This distinguishes the category Ste from the
other known categories of locally convex spaces since up to the recent
time only the category Ban of Banach spaces and the category Fin of
finite-dimensional spaces had been known to possess this property. On
the other hand, the category Ste is so wide, and the tools for
creating new spaces in Ste are so diverse, that this suggests the idea
that all the results of functional analysis can be reformulated inside
the stereotype theory without essential losses. On this way one can
even try to completely replace the category of locally convex spaces
in analysis (and in related areas) by the category Ste of stereotype
spaces with the view of possible simplifications – this program was
announced by S. Akbarov in 2005[5] and the following results can be
considered as evidence of its reasonableness: In the theory of
stereotype spaces, the approximation property is inherited by the
spaces of operators and by tensor products. This allows to reduce the
list of counterexamples in comparison with the Banach theory, where as
is known the space of operators does not inherit the approximation
property.[63] The arising theory of stereotype algebras allows to
simplify constructions in the duality theories for non-commutative
groups. In particular, the group algebras (and their envelopes in the
necessary cases) in these theories become Hopf algebras in the
standard algebraic sense.[64][65][66][24] This in its turn leads to a
family of generalizations of the Pontryagin duality based on the
notion of envelope: the holomorphic, the smooth and the continuous
envelopes of stereotype algebras give rise respectively to the
holomorphic, the smooth and the continuous dualities in big geometric
disciplines – complex geometry, differential geometry, and topology
– for certain classes of (not necessarily commutative) topological
groups considered in these disciplines (affine algebraic groups, and
some classes of Lie groups and Moore groups).[22][23][24][66] See
also[edit] Stereotype algebra Stereotype group algebra Reflexive space
Duality (mathematics) Pontryagin duality Envelope (category theory)
Refinement (category theory) Brauner space Smith space Notes[edit] ↑
Akbarov 2003, p. 219. ↑ ...or over the field [math]\displaystyle{
\mathbb{R} }[/math] of real numbers, with the similar definition. ↑
Akbarov 2003, p. 219, 220. ↑ A set [math]\displaystyle{ D\subseteq
X }[/math] is said to be capacious if for each totally bounded set
[math]\displaystyle{ A\subseteq X }[/math] there is a finite set
[math]\displaystyle{ F\subseteq X }[/math] such that
[math]\displaystyle{ A\subseteq D+F }[/math]. ↑ 5.0 5.1 Akbarov
2005. ↑ Akbarov 2003, p. 220, Example 4.3. ↑ Of course, this is a
general fact: if [math]\displaystyle{ X }[/math] is stereotype then
[math]\displaystyle{ X^\star }[/math] is also stereotype. ↑ Akbarov
2003, p. 221, Example 4.8. ↑ Akbarov 2003, p. 221, Theorem 4.11.
↑ A locally convex space [math]\displaystyle{ X }[/math] is called
co-complete if each linear functional [math]\displaystyle{
f:X\to\mathbb{C} }[/math] which is continuous on every totally bounded
set [math]\displaystyle{ S\subseteq X }[/math], is automatically
continuous on the whole space [math]\displaystyle{ X }[/math]. ↑ A
locally convex space [math]\displaystyle{ X }[/math] is said to be
saturated if for an absolutely convex set [math]\displaystyle{
B\subseteq X }[/math] being a neighbourhood of zero in
[math]\displaystyle{ X }[/math] is equivalent to the following: for
each totally bounded set [math]\displaystyle{ S\subseteq X }[/math]
there is a closed neighbourhood of zero [math]\displaystyle{ U
}[/math] in [math]\displaystyle{ X }[/math] such that
[math]\displaystyle{ B\cap S=U }[/math]. ↑ A locally convex space
[math]\displaystyle{ X }[/math] is called a Pták space, or a fully
complete space, if in its dual space [math]\displaystyle{ X^\star
}[/math] a subspace [math]\displaystyle{ Q\subseteq X^\star }[/math]
is [math]\displaystyle{ X }[/math]-weakly closed when it has
[math]\displaystyle{ X }[/math]-weakly closed intersection with the
polar [math]\displaystyle{ U^\circ }[/math] of each neighbourhood of
zero [math]\displaystyle{ U\subseteq X }[/math]. ↑ A locally convex
space [math]\displaystyle{ X }[/math] is said to be hypercomplete if
in its dual space [math]\displaystyle{ X^\star }[/math] every
absolutely convex space [math]\displaystyle{ Q\subseteq X^\star
}[/math] is [math]\displaystyle{ X }[/math]-weakly closed if it has
[math]\displaystyle{ X }[/math]-weakly closed intersection with the
polar [math]\displaystyle{ U^\circ }[/math] of each neighbourhood of
zero [math]\displaystyle{ U\subseteq X }[/math]. ↑ Akbarov 2003,
p. 221, Example 4.10. ↑ Akbarov 2003, p. 221, Example 4.9. ↑
Smith 1952. ↑ Onishchik 1984. ↑ Brudovski 1967. ↑ Waterhouse
1968. ↑ Brauner 1973. ↑ Akbarov 2003. ↑ 22.0 22.1 Akbarov 2009.
↑ 23.0 23.1 Akbarov 2016. ↑ 24.0 24.1 24.2 Akbarov 2017. ↑
Akbarov & Shavgulidze 2003. ↑ Akbarov 1995. ↑ Akbarov 2003,
p. 197. ↑ Akbarov 2022, Theorem 3.3.10. ↑ Akbarov 2003, p. 200.
↑ Akbarov 2022, Theorem 3.3.24. ↑ It is not clear (2017) whether
[math]\displaystyle{ X^{\vartriangle\triangledown} }[/math] and
[math]\displaystyle{ X^{\triangledown\vartriangle} }[/math] coincide.
↑ A monomorphism [math]\displaystyle{ \mu }[/math] is said to be
immediate if in each representation [math]\displaystyle{
\mu=\mu'\circ\varepsilon }[/math], where [math]\displaystyle{ \mu'
}[/math] is a monomorphism and [math]\displaystyle{ \varepsilon
}[/math] is an epimorphism, the morphism [math]\displaystyle{
\varepsilon }[/math] is automatically an isomorphism. ↑ 33.0 33.1
Akbarov 2016, p. 39. ↑ A monomorphism [math]\displaystyle{ \mu:C\to
D }[/math] is said to be strong, if for any epimorphism
[math]\displaystyle{ \varepsilon:A\to B }[/math] and for any morphisms
[math]\displaystyle{ \alpha:A\to C }[/math] and [math]\displaystyle{
\beta:B\to D }[/math] such that [math]\displaystyle{
\beta\circ\varepsilon=\mu\circ\alpha }[/math] there exists a morphism
[math]\displaystyle{ \delta:B\to C }[/math], such that
[math]\displaystyle{ \delta\circ\varepsilon=\alpha }[/math] and
[math]\displaystyle{ \mu\circ\delta=\beta }[/math]. ↑ In other
words, in this case the topology of [math]\displaystyle{ Y }[/math]
inherited from [math]\displaystyle{ X }[/math] is not pseudosaturated.
↑ 36.0 36.1 Akbarov 2016, p. 128. ↑ Akbarov 2016, p. 134. ↑
Akbarov 2016, p. 131. ↑ I.e. the linear span of
[math]\displaystyle{ M }[/math] is dense in [math]\displaystyle{
\operatorname{Env}^XM }[/math] (as in a locally convex space). ↑
40.0 40.1 Akbarov 2016, p. 144. ↑ An epimorphism
[math]\displaystyle{ \varepsilon }[/math] is said to be immediate if
in each representation [math]\displaystyle{
\varepsilon=\mu\circ\varepsilon' }[/math], where [math]\displaystyle{
\mu }[/math] is a monomorphism and [math]\displaystyle{ \varepsilon'
}[/math] is an epimorphism, the morphism [math]\displaystyle{ \mu
}[/math] is automatically an isomorphism. ↑ An epimorphism
[math]\displaystyle{ \varepsilon:A\to B }[/math] is said to be strong,
if for any monomorphism [math]\displaystyle{ \mu:C\to D }[/math] and
for any morphisms [math]\displaystyle{ \alpha:A\to C }[/math] and
[math]\displaystyle{ \beta:B\to D }[/math] such that
[math]\displaystyle{ \beta\circ\varepsilon=\mu\circ\alpha }[/math]
there exists a morphism [math]\displaystyle{ \delta:B\to C }[/math],
such that [math]\displaystyle{ \delta\circ\varepsilon=\alpha }[/math]
and [math]\displaystyle{ \mu\circ\delta=\beta }[/math]. ↑ A linear
map [math]\displaystyle{ \varphi:X\to Y }[/math] is said to be open,
if for each neighborhood of zero [math]\displaystyle{ U\subseteq X
}[/math] there is a neighborhood of zero [math]\displaystyle{
V\subseteq Y }[/math] such that [math]\displaystyle{
\varphi(U)\supseteq V\cap\varphi(X) }[/math]. ↑ 44.0 44.1 Akbarov
2016, p. 138. ↑ Akbarov 2016, p. 140. ↑ This proposition is dual
to the corresponding theorem on envelopes in stereotype spaces. ↑
I.e. if [math]\displaystyle{ y\ne z\in\operatorname{Ref}^XF }[/math],
then there exists [math]\displaystyle{ f\in F }[/math] such that
[math]\displaystyle{ f(y)\ne f(z) }[/math]. ↑ 48.0 48.1 Akbarov
2003, p. 224. ↑ Akbarov 2003, p. 226. ↑ Akbarov 2003, p. 220.
↑ Akbarov 2016, p. 142. ↑ 52.0 52.1 52.2 Akbarov 2003, p. 245.
↑ Akbarov 2009, p. 480-481. ↑ 54.0 54.1 Akbarov 2017, p. 581.
↑ Akbarov 2003, 7.17. ↑ Akbarov 2003, 7.21. ↑ 57.0 57.1 Akbarov
2003, (4.15). ↑ Akbarov 2003, p. 246. ↑ Akbarov 2003, p. 264.
↑ 60.0 60.1 Akbarov 2003, p. 265. ↑ Akbarov 2003, p. 242. ↑
Akbarov 2003, p. 289. ↑ Szankowski 1981. ↑ Akbarov 2003, p. 278.
↑ Akbarov 2009, p. 507. ↑ 66.0 66.1 Kuznetsova 2013. External
links[edit] Stereotype space in nLab References[edit] Akbarov, S.S.
(2022). Stereotype spaces and algebras. De Gruyter Expositions in
Mathematics. De Gruyter. ISBN 9783110780918.
https://www.degruyter.com/document/isbn/9783110780918/html. 
Onishchik, A.L. (1984). Pontrjagin duality. 4. 481–482.
ISBN 978-1402006098.
https://www.encyclopediaofmath.org/index.php/Pontryagin_duality. 
Smith, M.F. (1952). "The Pontrjagin duality theorem in linear spaces".
Annals of Mathematics 56 (2): 248–253. doi:10.2307/1969798. 
Brudovski, B.S. (1967). "On k- and c-reflexivity of locally convex
vector spaces". Lithuanian Mathematical Journal 7 (1): 17–21. 
Waterhouse, W.C. (1968). "Dual groups of vector spaces". Pac. J. Math.
26 (1): 193–196. doi:10.2140/pjm.1968.26.193.
http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pjm/1102986038&page=record. 
Brauner, K. (1973). "Duals of Fréchet spaces and a generalization of
the Banach–Dieudonné theorem". Duke Math. Jour. 40 (4): 845–855.
doi:10.1215/S0012-7094-73-04078-7.  Akbarov, S.S. (1995). "Pontryagin
duality in the theory of topological vector spaces". Mathematical
Notes 57 (3): 319–322. doi:10.1007/BF02303980.  Akbarov, S.S.
(2003). "Pontryagin duality in the theory of topological vector spaces
and in topological algebra". Journal of Mathematical Sciences 113 (2):
179–349. doi:10.1023/A:1020929201133.
http://www.springerlink.com/content/k62m72960101g6q2/.  Akbarov, S.S.
(2009). "Holomorphic functions of exponential type and duality for
Stein groups with algebraic connected component of identity". Journal
of Mathematical Sciences 162 (4): 459–586.
doi:10.1007/s10958-009-9646-1.  Akbarov, S.S. (2016). "Envelopes and
refinements in categories, with applications to functional analysis".
Dissertationes Mathematicae 513: 1–188. doi:10.4064/dm702-12-2015.
https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513. 
Akbarov, S.S.; Shavgulidze, E.T. (2003). "On two classes of spaces
reflexive in the sense of Pontryagin". Mat. Sbornik 194 (10): 3–26.
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=771&option_lang=eng. 
Akbarov, S.S. (2017). "Continuous and smooth envelopes of topological
algebras. Part 1". Journal of Mathematical Sciences 227 (5):
531–668. doi:10.1007/s10958-017-3599-6.  Akbarov, S.S. (2017).
"Continuous and smooth envelopes of topological algebras. Part 2".
Journal of Mathematical Sciences 227 (6): 669–789.
doi:10.1007/s10958-017-3600-4.  Kuznetsova, Y. (2013). "A duality for
Moore groups". Journal of Operator Theory 69 (2): 101–130.
doi:10.7900/jot.2011mar17.1920. Bibcode: 2009arXiv0907.1409K. 
Akbarov, S.S. (2005). "Pontryagin duality and topological algebras".
Banach Center Publications 67: 55–71. doi:10.4064/bc67-0-5.
http://webmail.impan.gov.pl/cgi-bin/bc/pdf?bc67-0-05.  Szankowski, A.
(1981). "B(H) does not have the approximation property". Acta Math.
147: 89–108. doi:10.1007/bf02392870.  vteFunctional
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Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach (hyperplane
separation) Kakutani fixed-point Krein–Milman Lomonosov's invariant
subspace Mackey–Arens Mazur's lemma M. Riesz extension Riesz
representation Parseval's identity Schauder fixed-point Analysis
Abstract Wiener space Bochner space Differentiation in Fréchet spaces
Derivatives (Fréchet Gateaux functional holomorphic)
Integrals (Bochner Dunford Gelfand–Pettis regulated Paley–Wiener
weak) Functional calculus (Borel continuous holomorphic) Inverse
function theorem (Nash–Moser theorem) Measures (Lebesgue
Projection-valued Vector) Weakly measurable function Types of sets
Absolutely convex Absorbing Balanced Bounded Convex Convex cone
(subset) Linear cone (subset) Radial Star-shaped Symmetric Zonotope
Subsets / set operations Algebraic interior (core) Bounding points
Convex hull Extreme point Interior Minkowski addition Polar
vteCategory theoryKey conceptsKey concepts Category Adjoint functors
Commutative diagram Concrete category Functor Morphism Natural
transformation Universal property Universal constructionsLimits
Terminal objects Products Equalizers Kernels Pullbacks Inverse limit
Colimits Initial objects Coproducts Coequalizers Cokernels and
quotients Pushout Direct limit Algebraic categories Sets Relations
Magmas Groups Abelian groups Rings (Fields) Modules (Vector spaces)
Constructions on categories Free category Functor category Quotient
category Product category Subcategory Higher category theory Key
concepts Categorification Enriched category Higher-dimensional algebra
Homotopy hypothesis Model category Simplex category n-categoriesWeak
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan
complex ∞-groupoid ∞-topos Strict n-categories 2-category
(2-functor) 3-category Categorified concepts 2-group 2-ring En-ring
(Symmetric) monoidal category n-group n-monoid Category Mathematics
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