Re: Tensor products of some Banach spaces

In article <3o0b3rF3eajqU2@xxxxxxxxxxxxxx>, José Carlos Santos
<jcsantos@xxxxxxxx> wrote:

> tommi.hoynalanmaa@xxxxxx wrote:
>
> > If we have two normed linear spaces E and F is there some standard way
> > to define norm on tensor product E X F (tensor product of two vector
> > spaces) so that E X F becomes a normed linear space?
>
> There are lots of ways. You can, for instance, define
>
> ||(v,w)|| = ||v|| + ||w||
>
> or
>
> ||(v,w)|| = max{||v||,||w||}.
>
> Best regards,
>
> Jose Carlos Santos

No, those are norms for the direct sum (= Cartesian product), not the
tensor product.

The theory of infinite-dimensional tensor products goes back to
Grothendieck. Based on the simple duality of tensor product with
bilinear forms.

There are multiple tensor norms used, but if E and F are complete, the
tensor product often isn't, so then you take the completion.
[Work on tensor norms is often phrased in a dual form in terms of
"operator ideals".]

See
http://en.wikipedia.org/wiki/Tensor_product
which unfortunately has only Tensor products of Hilbert spaces.

An Amazon.com search produces:

Introduction to Tensor Products of Banach Spaces
by Raymond A. Ryan

Functors and categories of Banach spaces: Tensor products, operator
ideals, and functors on categories of Banach spaces (Lecture notes in
mathematics ; 651)
by Peter W Michor

Tensor Norms and Operator Ideals (North-Holland Mathematics Studies)
by A. Defant, K. Floret

Operator ideals (Mathematische Monographien)
by A Pietsch

Grothendieck ideals of operators on Banach spaces
by H. P Lotz

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
.

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