2
votes
0answers
103 views

Projective tensor powers of Banach spaces over a normed field

Let $E$ be a Banach space over a complete normed field $\mathbb K$. Is it possible to classify all proper $E$ for which the projective seminorm $p_n$ defined on the $n$-th tensor power of $E$ is a ...
3
votes
1answer
170 views

Does the topological Varopoulos algebra consist of functions that are continuous and Varopoulos norm bounded?

Let $X_1,\dots,X_n$ be compact Hausdorff spaces. Let's define the Varopoulos algebra as the projective tensor product: $$V(X_1,\dots,X_n) := C(X_1) \hat{\otimes} \dots \hat{\otimes} C(X_n),$$ i.e. the ...
1
vote
0answers
179 views

Tensors with low spectral norm

Consider a tensor $T$ with six indices, $T_{(ii')(jj')(kk')}$, where each index goes from $1$ to $n$. We can think of $T$ as a linear map from $\mathbb{R}^n \otimes \mathbb{R}^n \otimes \mathbb{R}^n$ ...
3
votes
1answer
335 views

injective tensor norms for real tensors

If $A$ is an element of $\mathbb{R}^n \otimes\mathbb{R}^n \otimes\mathbb{R}^n$, then define its injective tensor norm to be $$\|A\|_{\rm inj} := \max_{x,y,z\in \mathbb{R}^n, \|x\|=\|y\|=\|z\|=1} ...
3
votes
0answers
1k views

Projective and injective tensor product

It is well known that for arbitrary Banach spaces $X$ and $Y$ we have that the dual space $(X \hat{\otimes}_{\pi} Y)^* = \mathcal{L}(X, Y^*)$. If we take $\ell^p$ and $\ell^q$ such that $p < ...
4
votes
1answer
466 views

What is the tensor product of $L^p(\bf R)$ with $L^q(\bf R)$?

I'm wondering: What is the tensor product of $L^p({\bf R})$ with $L^q({\bf R})$? (For p=q=2, the answer clearly should be $L^2({\bf R}^2)$; for other values of $p$ and $q$, it is not at all obvious ...