Constructing Norms on Tensor Products of Finite Dimensional Vector Spaces

[Pere Callahan]

I was wondering about useful norms on tensor products of finite dimensional vector spaces.

Let V,W be two such vector spaces with bases $\{v_1,\ldots,v_{d_1}\}$ and $\{w_1,\ldots,w_{d_2}\}$. We further assume that each is equipped with a norm, $||\cdot||_V$ and $||\cdot||_W$.

Then the tensor product space $V\otimes W$ is the vector space with basis $\{v_i\otimes w_j:1\leq i\leq d_1, 1\leq j\leq d_2\}$.

I have read a lot about norming the tensor product of two Banach spaces and there a lot of different choices can be made it seems. For the finite dimensional case I would be interested to know how one defines a norm $\|\cdot\|_\otimes$ on $V\otimes W$ such that for pure tensors it holds that $\|v\otimes w\|_\otimes=\|u\|_V\|w\|_W$. Such norm are called crossnorms in the Banach space context (or so it seems) but I have not seen a construction of such a norm even for the finite dimensional case.

This is no homework.

Thanks,
PereEDIT:

It appears that if $\|\sum_{i=1}^{d_1}{x_iv_i}\|_V$ is defined as $\sum_{i=1}^{d_1}{|x_i|}$ and similarly for W then one could define
$$\|\sum_{i,j}\gamma_{ij}v_i\otimes w_j\|_\otimes = \sum_{i,j}{|\gamma_{ij}|}.$$
This would be a norm and would satisfy the crossnorm condition because
$$\left\|v\otimes w\right\|_\otimes=\left\|\left(\sum_{i=1}^{d_1}x_iv_i\right)\otimes\left(\sum_{j=1}^{d_2}y_jw_j\right)\right\|_\otimes = \left\|\sum_{ij}x_iy_j v_i\otimes w_j\right\|_\otimes = \sum_{ij}|x_i||y_j|=\|v\|_V\|w\|_W.$$

However there should be a more general construction for arbitrary norms on V and W.

 

[hamster143]

This is late so I may be wrong, but here's what I think.

In the general case, you can take

$$\sum_{i,j}\gamma_{ij}v_i\otimes w_j$$

and apply singular value decomposition to $\gamma$, thus rewriting your element in the form

$$\sum_i \gamma_i v'_i \otimes w'_i$$

such that

$$\|v'_i\|_V = \|w'_i\|_W = 1$$

and then use any convex function $F(\gamma_i)$ such that $F(1,0,...0)=1$, ..., $F(\alpha \gamma_i) = |\alpha| F(\gamma_i)$to define

$$\|\sum_{i,j}\gamma_{ij}v_i\otimes w_j\|_\otimes = F(\gamma_i).$$

Crossnorm condition is satisfied by construction. It is a bit tricky to prove that it is a norm, triangle inequality looks particularly challenging ... Maybe try special cases $F(\gamma_i) = \sum_i |\gamma_i|$ and $F(\gamma_i) = \sqrt{\sum_i \gamma_i^2}$.

 

[hamster143]

On further thought, I began to doubt that it's a norm for any, let alone all F. Besides, for some norms on V and W, it may not be unique.

Oh well, I don't know then...

 

[Pere Callahan]

Thanks for your thought hamster. I also don't see how the general construction with the SVD and a convex function F would give a norm. Well, the example I gave in my first post certainly extends to all p-norms on V and W and maybe that's enough.

 

[hamster143]

Have you seen this;

http://en.wikipedia.org/wiki/Topological_tensor_product

There is a largest cross norm π called the projective cross norm, given by

$$\pi(x) = \inf \{ \Sigma_{i=1}^n \|a_i\| \|b_i\| : x = \Sigma a_i \otimes b_i\}$$

where $x \in A \otimes B.$

 

[Pere Callahan]

Yes I knew about the projective and injective tensor norms. I just found them little intuitive and was hoping for more explicit constructions in the finite dimensional case. But I think for my purposes the p-norms will do.