Difference between revisions of "Topological tensor product"
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− | ''of two locally convex spaces | + | ''of two locally convex spaces $E_1$ and $E_2$'' |
− | A [[Locally convex space|locally convex space]] having a universality property with respect to bilinear operators on | + | A [[Locally convex space|locally convex space]] having a universality property with respect to bilinear operators on $E_1 \times E_2$ and satisfying a continuity condition. More precisely, let $\mathcal{K}$ be a certain class of locally convex spaces and for each $F \in \mathcal{K}$ let there be given a subset $T(F)$ of the set of separately-continuous bilinear operators from $E_1 \times E_2$ into $F$. Then the topological tensor product of $E_1$ and $E_2$ (with respect to $T(F)$) is the (unique) locally convex space $E_1 \tilde\otimes E_2 \in \mathcal{K}$ together with the operator $B \in T(E_1 \tilde\otimes E_2)$ having the following property: For any $S \in T(F)$, $F \in \mathcal{K}$, there exists a unique continuous linear operator $R:E_1 \tilde\otimes E_2 \to F$ such that $R \circ B = S$. Thus, if one speaks of the functor $T: \mathcal{} \to \mathrm{Sets}$, then $E_1 \tilde\otimes E_2$ is defined as the representing object of this functor. |
In all known examples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316020.png" /> contains the field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316021.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316022.png" /> contains all bilinear functionals of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316025.png" />, mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316026.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316027.png" />. If in this case the topological tensor product exists, then there is a dense subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316028.png" /> that can be identified with the algebraic [[Tensor product|tensor product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316029.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316030.png" />. | In all known examples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316020.png" /> contains the field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316021.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316022.png" /> contains all bilinear functionals of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316025.png" />, mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316026.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316027.png" />. If in this case the topological tensor product exists, then there is a dense subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316028.png" /> that can be identified with the algebraic [[Tensor product|tensor product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316029.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316030.png" />. |
Revision as of 22:09, 9 April 2019
of two locally convex spaces $E_1$ and $E_2$
A locally convex space having a universality property with respect to bilinear operators on $E_1 \times E_2$ and satisfying a continuity condition. More precisely, let $\mathcal{K}$ be a certain class of locally convex spaces and for each $F \in \mathcal{K}$ let there be given a subset $T(F)$ of the set of separately-continuous bilinear operators from $E_1 \times E_2$ into $F$. Then the topological tensor product of $E_1$ and $E_2$ (with respect to $T(F)$) is the (unique) locally convex space $E_1 \tilde\otimes E_2 \in \mathcal{K}$ together with the operator $B \in T(E_1 \tilde\otimes E_2)$ having the following property: For any $S \in T(F)$, $F \in \mathcal{K}$, there exists a unique continuous linear operator $R:E_1 \tilde\otimes E_2 \to F$ such that $R \circ B = S$. Thus, if one speaks of the functor $T: \mathcal{} \to \mathrm{Sets}$, then $E_1 \tilde\otimes E_2$ is defined as the representing object of this functor.
In all known examples contains the field of complex numbers
, and
contains all bilinear functionals of the form
,
,
, mapping
to
. If in this case the topological tensor product exists, then there is a dense subspace in
that can be identified with the algebraic tensor product
; moreover,
.
If consists of all separately (respectively, jointly) continuous bilinear operators, then the topological tensor product is called inductive (respectively, projective). The most important is the projective topological tensor product. Let
be a defining family of semi-norms in
,
; denote by
the topology on
defined by the family of semi-norms
:
![]() |
If is the class of all, respectively all complete, locally convex spaces, then the projective topological tensor product of
and
exists and its locally convex space is
with the topology
, respectively its completion. If the
are Banach spaces with norms
,
, then
is a norm on
; the completion with respect to it is denoted by
. For each
the elements of
have the representation
![]() |
where
![]() |
If one endows with a topology weaker than
by using the family of semi-norms
,
![]() |
where and
are the polar sets of the unit spheres with respect to
and
, then there arises a topological tensor product, sometimes called injective. The locally convex spaces
with the property that for an arbitrary
both topologies in
coincide, form the important class of nuclear spaces (cf. Nuclear space).
The projective topological tensor product is associated with the approximation property: A locally convex space has the approximation property if for each pre-compact set
and neighbourhood of zero
there exists a continuous operator of finite rank
such that for all
one has
. All nuclear spaces have the approximation property. A Banach space
has the approximation property if and only if for an arbitrary Banach space
the operator
, unambiguously defined by the equation
, has trivial kernel. A separable Banach space without the approximation property has been constructed [3]. This space also gives an example of a Banach space without a Schauder basis, since the Banach spaces with a Schauder basis have the approximation property (thus S. Banach's so-called "Banach basis problembasis problem" has been negatively solved).
References
[1] | A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc. (1955) |
[2] | H.H. Schaefer, "Topological vector spaces" , Macmillan (1966) |
[3] | P. Enflo, "A counterexample to the approximation problem in Banach spaces" Acta Math. , 130 (1973) pp. 309–317 |
Comments
References
[a1] | A. Pietsch, "Nukleare lokalkonvexe Räume" , Akademie Verlag (1965) |
[a2] | J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1. Sequence spaces , Springer (1977) |
[a3] | F. Trèves, "Topological vectorspaces, distributions and kernels" , Acad. Press (1967) pp. 198 |
[a4] | G. Pisier, "Factorisation of linear operators and geometry of Banach spaces" , Amer. Math. Soc. (1986) |
Topological tensor product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_tensor_product&oldid=12428