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Norm on tensor product

  1. Jun 27, 2010 #1
    I was wondering about useful norms on tensor products of finite dimensional vector spaces.

    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.
     
    Last edited: Jun 27, 2010
  2. jcsd
  3. Jun 29, 2010 #2
    This is late so I may be wrong, but here's what I think.

    In the general case, you can take

    [tex]\sum_{i,j}\gamma_{ij}v_i\otimes w_j[/tex]

    and apply singular value decomposition to [itex]\gamma[/itex], thus rewriting your element in the form

    [tex] \sum_i \gamma_i v'_i \otimes w'_i[/tex]

    such that

    [tex]\|v'_i\|_V = \|w'_i\|_W = 1[/tex]

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

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

    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 [itex]F(\gamma_i) = \sum_i |\gamma_i|[/itex] and [itex]F(\gamma_i) = \sqrt{\sum_i \gamma_i^2}[/itex].
     
    Last edited: Jun 29, 2010
  4. Jun 29, 2010 #3
    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...
     
  5. Jun 29, 2010 #4
    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.
     
  6. Jun 29, 2010 #5
    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

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

    where [itex]x \in A \otimes B.[/itex]
     
  7. Jun 29, 2010 #6
    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.
     
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