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 closed monoidal category with the standard analytical tools for constructing new spaces, like taking closed subspace, quotient space, projective and injective limits, the space of operators, tensor products, etc.

Mutual embeddings of the main classes of locally convex spaces

Definition

A stereotype space^[1] is a topological vector space $X$ over the field $\mathbb{C}$ of complex numbers^[2] such that the natural map into the second dual space

$i:X\to X^{\star\star},\quad i(x)(f)=f(x),\quad x\in X,\quad f\in X^\star$

is an isomorphism of topological vector spaces (i.e. a linear and a homeomorphic map). Here the dual space $X^\star$ is defined as the space of all linear continuous functionals $f:X\to\mathbb{C}$ endowed with the topology of uniform convergence on totally bounded sets in X, and the second dual space $X^{\star\star}$ is the space dual to $X^{\star}$ in the same sense.

The following criterion holds:^[1] a topological vector space $X$ is stereotype if and only if it is locally convex and satisfies the following two conditions:

• pseudocompleteness: each totally bounded Cauchy net in $X$ converges,
• pseudosaturateness: each closed convex balanced capacious^[3] set $D$ in $X$ is a neighborhood of zero in $X$.

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

Each pseudocomplete barreled space $X$ (in particular, each Banach space and each Fréchet space) is stereotype. A metrizable locally convex space $X$ is stereotype if and only if $X$ is complete. Each infinite dimensional normed space $X$ considered with the $X^{\star}$-weak topology is not stereotype. There exist stereotype spaces which are not Mackey spaces.

Some simple connections between the properties of a stereotype space $X$ and those of its dual space $X^\star$ are expressed in the following list of regularities.^[1][4] For a stereotype space $X$

• $X$ is a normed space $\Longleftrightarrow$ $X$ is a Banach space $\Longleftrightarrow$ $X^\star$ is a Smith space;

• $X$ is metrizable $\Longleftrightarrow$ $X$ is a Fréchet space $\Longleftrightarrow$ $X^\star$ is a Brauner space;

• $X$ is barreled $\Longleftrightarrow$ $X^\star$ has the Heine-Borel property;

• $X$ is quasibarreled $\Longleftrightarrow$ in $X^\star$ if a set $T$ is absorbed by each barrel, then $T$ is totally bounded;

• $X$ is a Mackey space $\Longleftrightarrow$ in $X^\star$ every $X$-weakly compact set is compact;

• $X$ is a Montel space $\Longleftrightarrow$ $X$ is barreled and has the Heine-Borel peoperty $\Longleftrightarrow$ $X^\star$ is a Montel space;

• $X$ is a space with a weak topology $\Longleftrightarrow$ in $X^\star$ every compact set $T$ is finite-dimensional;

• $X$ is separable $\Longleftrightarrow$ in $X^\star$ there is a sequence of closed subspaces $L_n$ of finite co-dimension with trivial intersection: $\bigcap_{n=1}^{\infty}L_n=\{0\}$.

• $X$ has the (classical) approximation property $\Longleftrightarrow$ $X^\star$ has the (classical) approximation property;

• $X$ is complete $\Longleftrightarrow$ $X^\star$ is co-complete^[5] $\Longleftrightarrow$ $X^\star$ is saturated;^[6]

• $X$ is a Pták space^[7] $\Longleftrightarrow$ in $X^\star$ a subspace $L$ is closed if it has the closed intersection $L\cap K$ with each compact set $K\subseteq X^\star$;

• $X$ is hypercomplete^[8] $\Longleftrightarrow$ in $X^\star$ an absolutely convex set $B$ is closed if it has the closed intersection $B\cap K$ with each compact set $K\subseteq X^\star$.

History

The first results on this type of reflexivity of topological vector spaces were obtained by M. F. Smith^[9] in 1952. Further investigations were conducted by B. S. Brudovskii, ^[10] W. C. Waterhouse,^[11] K. Brauner,^[12] S. S. Akbarov,^[1][4][13] and E. T. Shavgulidze.^[14]

Pseudocompletion and pseudosaturation

Each locally convex space $X$ can be transformed into a stereotype space with the help of the standard operations of pseudocompletion and pseudosaturation defined by the following two propositions.^[1]

1. With any locally convex space $X$, one can associate a linear continuous map $\triangledown_X: X\to X^\triangledown$ into some pseudocomplete locally convex space $X^\triangledown$, called pseudocompletion of $X$, in such a way that the following conditions are fulfilled:

• $X$ is pseudocomplete if and only if $\triangledown_X: X\to X^\triangledown$ is an isomorphism;
• for any linear continuous map $\varphi:X\to Y$ of locally convex spaces, there exists a unique linear continuous map $\varphi^\triangledown:X^\triangledown\to Y^\triangledown$ such that $\triangledown_Y\circ\varphi=\varphi^\triangledown\circ\triangledown_X$.

One can imagine the pseudocompletion of $X$ as the "nearest to $X$ from the outside" pseudocomplete locally convex space, so that the operation $X\mapsto X^\triangledown$ adds to $X$ some supplementary elements, but does not change the topology of $X$ (like the usual operation of completion).

2. With any locally convex space $X$, one can associate a linear continuous map $\vartriangle_X:X^\vartriangle\to X$ from some pseudosaturated locally convex space $X^\vartriangle$, called pseudosaturation of $X$, in such a way that the following conditions are fulfilled:

• $X$ is pseudosaturated if and only if $\vartriangle_X:X^\vartriangle\to X$ is an isomorphism;
• for any linear continuous map $\varphi:X\to Y$ of locally convex spaces, there exists a unique linear continuous map $\varphi^\vartriangle:X^\vartriangle\to Y^\vartriangle$ such that $\varphi\circ\vartriangle_X=\vartriangle_Y\circ\varphi^\vartriangle$.

The pseudosaturation of $X$ can be imagined as the "nearest to $X$ from the inside" pseudosaturated locally convex space, so that the operation $X\mapsto X^\vartriangle$ strengthen the topology of $X$, but does not change the elements of $X$.

If $X$ is a pseudocomplete locally convex space, then its pseudosaturation $X^\vartriangle$ is stereotype. Dually, if $X$ is a pseudosaturated locally convex space, then its pseudocompletion $X^\triangledown$ is stereotype. For arbitrary locally convex space $X$ the spaces $X^{\vartriangle\triangledown}$ and $X^{\triangledown\vartriangle}$ are stereotype.^[15]

Category of stereotype spaces

The class Ste of stereotype spaces forms a category with linear continuous maps as morphisms and has the following properties:,^[1][13]

• Ste is pre-abelian;
• Ste is complete and co-complete;
• Ste is autodual with respect to the functor $X\to X^\star$ of passing to the dual space;
• Ste is a category with nodal decomposition: each morphism $\varphi:X\to Y$ has a decomposition $\varphi=\sigma\circ\beta\circ\pi$, where $\pi$ is a strong epimorphism, $\beta$ a bimorphism, and $\sigma$ a strong monomorphism.

For any two stereotype spaces $X$ and $Y$ the stereotype space of operators $\text{Hom}(X,Y)$ from $X$ into $Y$, is defined as the pseudosaturation of the space $\text{L}(X,Y)$ of all linear continuous maps $\varphi:X\to Y$ endowed with the topology of uniform convergeance on totally bounded sets. The space $\text{Hom}(X,Y)$ is stereotype. It defines two natural tensor products

$X\circledast Y:= \text{Hom}(X,Y^\star)^\star,$

$X\odot Y := \text{Hom}(X^\star,Y).$

The following natural identities hold:^[1]

$\mathbb{C}\circledast X\cong X\cong X\circledast \mathbb{C},$

$\mathbb{C}\odot X\cong X\cong X\odot\mathbb{C},$

$X\circledast Y\cong Y\circledast X,$

$X\odot Y\cong Y\odot X,$

$(X\circledast Y)\circledast Z\cong X\circledast (Y\circledast Z),$

$(X\odot Y)\odot Z\cong X\odot (Y\odot Z),$

$(X\circledast Y)^\star\cong Y^\star\odot X^\star,$

$(X\odot Y)^\star\cong Y^\star\circledast X^\star,$

$\text{Hom}(X\circledast Y,Z)\cong \text{Hom}(X,\text{Hom}(Y,Z)),$

$\text{Hom}(X,Y\odot Z)\cong \text{Hom}(X,Y)\odot Z$

As a corollary,

• Ste is a symmetric monoidal category with respect to the bifunctor $\odot$ and a symmetric closed monoidal category with respect to the bifunctor $\circledast$ and the internal hom-functor $\text{Hom}$.

Stereotype approximation property

A stereotype space $X$ is said to have the stereotype approximation property, if each linear continuous map $\varphi:X\to X$ can be approximated in the stereotype space of operators $\text{Hom}(X,X)$ 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 (2013) whether the stereotype approximation property coincide with the classical one, or not). The following proposition holds:

• If two stereotype spaces $X$ and $Y$ have the stereotype approximation property, then the spaces $\text{Hom}(X,Y)$, $X\circledast Y$ and $X\odot Y$ have the stereotype approximation property as well.^[1]

In particular, if $X$ has the stereotype approximation property, then the same is true for $X^\star$ and for $\text{Hom}(X,X)$.

Applications

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 for each stereotype algebra $A$ the categories _$A$Ste and Ste_$A$ of left and right stereotype modules over $A$ are enriched categories over Ste.^[1] 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 functional 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^[16] and the following results can be considered as evidences 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.^[17]
• The arising theory of stereotype algebras allows to simplify constructions in the duality theories for non-commutative groups. In particular, the group algebras in these theories become Hopf algebras in the standard algebraic sense.^[4][18]

Notes

1. S.S.Akbarov (2003).
2. ...or over the field $\mathbb{R}$ of real numbers, with the similar definition.
3. A set $D\subseteq X$ is said to be capacious if for each totally bounded set $A\subseteq X$ there is a finite set $F\subseteq X$ such that $A\subseteq D+F$.
4. S.S.Akbarov (2009).
5. A locally convex space $X$ is called co-complete if each linear functional $f:X\to\mathbb{C}$ which is continuous on every totally bounded set $S\subseteq X$, is automatically continuous on the whole space $X$.
6. A locally convex space $X$ is said to be saturated if for an absolutely convex set $B\subseteq X$ being a neighbourhood of zero in $X$ is equivalent to the following: for each totally bounded set $S\subseteq X$ there is a closed neighbourhood of zero $U$ in $X$ such that $B\cap S=U$.
7. A locally convex space $X$ is called a Pták space, or a fully complete space, if in its dual space $X^\star$ a subspace $Q\subseteq X^\star$ is $X$-weakly closed when it has $X$-weakly closed intersection with the polar $U^\circ$ of each neighbourhood of zero $U\subseteq X$.
8. A locally convex space $X$ is said to be hypercomplete if in its dual space $X^\star$ every absolutely convex space $Q\subseteq X^\star$ is $X$-weakly closed if it has $X$-weakly closed intersection with the polar $U^\circ$ of each neighbourhood of zero $U\subseteq X$.
9. M.F.Smith (1952).
10. B.S.Brudowski (1967).
11. W.C.Waterhouse (1968).
12. K.Brauner (1973).
13. S.S.Akbarov (2013).
14. S.S.Akbarov, E.T.Shavgulidze (2003).
15. It is not clear (2013) whether $X^{\vartriangle\triangledown}$ and $X^{\triangledown\vartriangle}$ coincide.
16. S.S.Akbarov (2005).
17. A.Szankowski (1981).
18. J.Kuznetsova (2013).

References

• Schaefer, Helmuth H. (1966). Topological vector spaces. New York: The MacMillan Company. ISBN 0-387-98726-6.

• Robertson, A.P.; Robertson, W.J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics 53. Cambridge University Press.

• Smith, M.F. (1952). "The Pontrjagin duality theorem in linear spaces". Annals of Mathematics 56 (2): 248–253. doi:10.2307/1969798. JSTOR 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.

• 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. (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.

• 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. (2013). "Envelopes and refinements in categories, with applications to functional analysis". arXiv:1110.2013 [math.FA].

• Akbarov, S.S.; Shavgulidze, E.T. (2003). "On two classes of spaces reflexive in the sense of Pontryagin". Mat. Sbornik 194 (10): 3–26.

• Kuznetsova, J. (2013). "A duality for Moore groups". Journal of Operator Theory 69 (2): 101–130. arXiv:0907.1409. Bibcode:2009arXiv0907.1409K. doi:10.7900/jot.2011mar17.1920.

• Akbarov, S.S. (2005). "Pontryagin duality and topological algebras". Banach Center Publications 67: 55–71. doi:10.4064/bc67-0-5.

• Szankowski, A. (1981). "B(H) does not have the approximation property". Act. Math. 147: 147:89–108. doi:10.1007/bf02392870.