2
votes
1answer
37 views

how do I view the tensor product $X^*\otimes Y$ as a subspace of $\mathcal{L}(X,Y)$?

Background. According to Raymond A. Ryan, in his book Introduction to Tensor Products of Banach Spaces, a Banach ideal $\mathcal{J}$ is an assignment to each pair of Banach spaces $X$ and $Y$ a ...
1
vote
1answer
24 views

Projections in Tensor Product

Let $X$ and $Y$ be Banach spaces (algebras). If $X\otimes^\pi Y$ denotes the projective tensor product of $X$and $Y$, define $P:X\otimes^\pi Y\rightarrow X$ as follows : for $x\otimes y\in X\otimes ...
1
vote
0answers
28 views

Simple tensors in the dual space

Let $X$ and $Y$ be two Banach spaces and assume, if necessary, that $X^*, Y^*$ have the approximation property (but not necessarily the Radon–Nikodym property). Consider the injective tensor product ...
2
votes
1answer
50 views

Dual of injective tensor norm is not projective tensor norm

Let $A$, $B$ are two Banach space, on the algebraic tensor space $A$ $\odot$ $B$, we can define the projection(maximal) tensor norm $\gamma$ and injective(minimal) tensor norm $\lambda$. For the ...
1
vote
1answer
43 views

The space of all bounded sequences over a Banach Algebra.

If $A$ is a commutative Banach algebra, can we identify $l^\infty(A)$ as the dual of $ l_1\hat\otimes A$ (the projective tensor product of $l_1$ with $A$? Also, what do we know about the quotient ...
0
votes
0answers
43 views

Largest/smallest cross-norm: A simple question about cross-norms on tensor products of Banach spaces.

This is a very simple dumb question as I’m completely new to the topic. I was reading Wikipedia’s entry on “Topological Tensor Products”, and there’s one thing I’m confused about. Let $ A $ and $ B $ ...
0
votes
0answers
34 views

Injective norm on tensor algebra of a finite-dimensional Banach space

Suppose that $V$ is a finite n-dimensional Banach space, and suppose that $T(V)$ is the tensor algebra on it. Furthermore, suppose that $T^{(n)}(V)$ is the "abridged" tensor algebra obtained by taking ...
2
votes
1answer
89 views

Ideals and exactness of projective tensor product of Banach spaces / algebras

Thanks for suggesting this question: Image of the tensor product of strict maps of Banach spaces I read the reference and realize that for a short exact sequence of Banach algebra: $0 \to J \to A \to ...
2
votes
0answers
62 views

Tensor products of weakly compact sets

Let $X$ and $Y$ be Banach spaces. Denote by $X\otimes_\varepsilon Y$ the injective tensor product of $X$ and $Y$. Also, let $A\subset X, B\subset Y$ be any sets. Set $A\otimes B = \{a\otimes b\colon ...
3
votes
1answer
65 views

Image of the tensor product of strict maps of Banach spaces

Let $f:A\to C$ and $g:B\to D$ be bounded linear maps of Banach spaces with closed image. Will $f\widehat{\otimes}g:A\widehat{\otimes}_\pi B\to C\widehat{\otimes}_\pi D$ also have closed image? What ...
2
votes
1answer
78 views

Calculating the norm of $S\otimes T$ for bounded linear $S,T$.

Let $X,Y,W,Z$ be Banach Spaces. Let $X\otimes_{\pi}Y$ denote the tensor product endowed with the projective norm. If I have $S\in B(X,Z)$, $T\in B(Y,W)$, it is straightforward to show that $S\otimes ...
5
votes
1answer
541 views

Proving that Tensor Product is Associative

I want to show that $X\otimes(Y\otimes Z)$ is isomorphic to $(X\otimes Y)\otimes Z$. Intuitively I think I should just choose bases $\{e_{i}\}_{i\in I}, \{f_{j}\}_{j\in J}$, and $\{g_{k}\}_{k\in K}$ ...
2
votes
1answer
78 views

Tensoring subspaces

Let $X$ be a Banach space, $E\subset X$, be a subspace and let $\hat{\otimes}$ denote the projective tensor product. Denote $L_1 = L_1 [0,1]$. Does $E\hat{\otimes} L_1$ embed into $X \hat{\otimes} ...
2
votes
1answer
184 views

How to describe the space $L_{\infty}(\mu,X)$?

Given a Banach space $X$ and a measure space $(\mathfrak{A}, \mu)$ One can form the Banach space $L_\infty(\mu, X)$ of all measurable, essentially bounded functions from $\mathfrak{A}$ to $X$. Is it ...
1
vote
0answers
111 views

Injective tensor products of WCG spaces

The notion of a WCG space is a common roof for separable Banach spaces and reflexive ones. Nevertheless, the class is stable under $\ell^p(\Gamma)$-sums for any set $\Gamma$ when $p>1$ and ...
35
votes
1answer
3k views

Was Grothendieck familiar with Stone's work on Boolean algebras?

In short, my question is: Was Grothendieck familiar with Stone's work on Boolean algebras? Background: In an answer to Pierre-Yves Gaillard's question Did Zariski really define the Zariski ...
4
votes
1answer
209 views

Tensor products and vector valued functions

Given a non-empty set $S$ and a Banach space $X$. Let $B(S,X)$ be the space of all bounded maps from $S$ to $X$. Can we identify $B(S,X)$ with $\ell^\infty(S) \otimes X$, where $\otimes$ is some kind ...
7
votes
3answers
1k views

Operator norm on product space

I have a bilinear operator $B\colon X \times Y\to Z$ with $X,Y,Z$ normed spaces, and define a norm on $X \times Y$ by $\lVert(x,y)\rVert = \lVert x\rVert_X + \lVert y\rVert_Y$ (using the respective ...
12
votes
1answer
1k views

Operator norm and tensor norms

I have a linear operator $A\in\mathcal{L}(X,Y)$ where $X$ and $Y$ are some Banach spaces (or Hilbert spaces would also do, if that simplifies the answer.). The operator norm of $A$ is given by $$ ...
7
votes
1answer
246 views

What is the ''right" norm for the Banach space tensor product in this situation?

Let $X,Y$ denote (real) vector spaces. The vector space of $n$-linear maps $X^n \to Y$ will be denoted by $L^n(X,Y)$. Unless I'm much mistaken $$L(X,L(X,Y)) \ \ \ L^2(X,Y) \ \ \ L(X \otimes X,Y)$$ ...
6
votes
3answers
274 views

$L^1$ space with values in a Banach Space

I have been reading a bit about the Bochner integral and now I'm wondering the following: For the theory to be "nice", one would expect that $$L^1([0, \tau], L^1([0, \tau])) \cong L^1([0,\tau] ...