View source for Topological tensor product

''of two locally convex spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t0931601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t0931602.png" />''

A [[Locally convex space|locally convex space]] having a universality property with respect to bilinear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t0931603.png" /> and satisfying a continuity condition. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t0931604.png" /> be a certain class of locally convex spaces and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t0931605.png" /> let there be given a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t0931606.png" /> of the set of separately-continuous bilinear operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t0931607.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t0931608.png" />. Then the topological tensor product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t0931609.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316010.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316011.png" />) is the (unique) locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316012.png" /> together with the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316013.png" /> having the following property: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316015.png" />, there exists a unique continuous linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316017.png" />. Thus, if one speaks of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316019.png" /> 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" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316031.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316032.png" /> be a defining family of semi-norms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316034.png" />; denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316035.png" /> the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316036.png" /> defined by the family of semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316037.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316038.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316039.png" /> is the class of all, respectively all complete, locally convex spaces, then the projective topological tensor product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316041.png" /> exists and its locally convex space is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316042.png" /> with the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316043.png" />, respectively its [[Completion|completion]]. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316044.png" /> are Banach spaces with norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316046.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316047.png" /> is a norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316048.png" />; the completion with respect to it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316049.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316050.png" /> the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316051.png" /> have the representation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316052.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316053.png" /></td> </tr></table>

If one endows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316054.png" /> with a topology weaker than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316055.png" /> by using the family of semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316056.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316057.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316059.png" /> are the polar sets of the unit spheres with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316061.png" />, then there arises a topological tensor product, sometimes called injective. The locally convex spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316062.png" /> with the property that for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316063.png" /> both topologies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316064.png" /> coincide, form the important class of nuclear spaces (cf. [[Nuclear space|Nuclear space]]).

The projective topological tensor product is associated with the approximation property: A locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316065.png" /> has the approximation property if for each pre-compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316066.png" /> and neighbourhood of zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316067.png" /> there exists a continuous operator of finite rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316068.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316069.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316070.png" />. All nuclear spaces have the approximation property. A Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316071.png" /> has the approximation property if and only if for an arbitrary Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316072.png" /> the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316073.png" />, unambiguously defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093160/t09316074.png" />, has trivial kernel. A separable Banach space without the approximation property has been constructed [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc.  (1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.H. Schaefer,   "Topological vector spaces" , Macmillan  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Enflo,   "A counterexample to the approximation problem in Banach spaces"  ''Acta Math.'' , '''130'''  (1973)  pp. 309–317</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Pietsch,   "Nukleare lokalkonvexe Räume" , Akademie Verlag  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Lindenstrauss,   L. Tzafriri,   "Classical Banach spaces" , '''1. Sequence spaces''' , Springer  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Trèves,   "Topological vectorspaces, distributions and kernels" , Acad. Press  (1967)  pp. 198</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Pisier,   "Factorisation of linear operators and geometry of Banach spaces" , Amer. Math. Soc.  (1986)</TD></TR></table>