Tagged Questions
1
vote
1answer
68 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
147 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
95 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 ...
33
votes
1answer
2k 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 ...
3
votes
1answer
185 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
632 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
630 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
$$
...
7
votes
1answer
206 views
What is the ''right" norm for the Banach space tensor product in this situation?
Let $X,Y$ denote (real) vector spaces. The vector space of $n$-linear maps $X^n \to Y$ will be denoted by $L^n(X,Y)$. Unless I'm much mistaken
$$L(X,L(X,Y)) \ \ \ L^2(X,Y) \ \ \ L(X \otimes X,Y)$$
...
6
votes
3answers
260 views
$L^1$ space with values in a Banach Space
I have been reading a bit about the Bochner integral and now I'm wondering the following:
For the theory to be "nice", one would expect that
$$L^1([0, \tau], L^1([0, \tau])) \cong L^1([0,\tau] ...
