I was wondering about useful norms on tensor products of finite dimensional vector spaces.(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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 [itex]\|\cdot\|_\otimes[/itex] on [itex]V\otimes W[/itex] such that for pure tensors it holds that [itex]\|v\otimes w\|_\otimes=\|u\|_V\|w\|_W[/itex]. 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,

Pere

EDIT:

It appears that if [itex]\|\sum_{i=1}^{d_1}{x_iv_i}\|_V[/itex] is defined as [itex]\sum_{i=1}^{d_1}{|x_i|}[/itex] and similarly for W then one could define

[tex]

\|\sum_{i,j}\gamma_{ij}v_i\otimes w_j\|_\otimes = \sum_{i,j}{|\gamma_{ij}|}.

[/tex]

This would be a norm and would satisfy the crossnorm condition because

[tex]

\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.

[/tex]

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

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# Norm on tensor product

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