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Question about linear mappings and inner product spaces

  1. Jul 18, 2014 #1
    1. The problem statement, all variables and given/known data

    Hi, I am having difficulty with the following proof:

    Let V be an inner product space (real of dimension n) with two inner products in V, <,> and [,]. Prove that there exists a linear mapping on V such that [L(x),L(y)] = <x,y> for all x,y in V.

    I am stuck as to where to go with the proof. I know that I need to construct a linear mapping with the above property, however I'm not sure where to go from there. Any insight into this would be appreciated.
    Thanks


    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jul 18, 2014 #2

    jbunniii

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    Hint: let ##\{e_1,\ldots,e_n\}## be an orthonormal (with respect to ##\langle , \rangle##) basis for ##V##. It suffices to find a linear map ##L## such that
    $$[L(e_i), L(e_j)] = \langle e_i, e_j\rangle = \delta_{ij}$$
    One way to proceed from here is to reformulate the problem in terms of matrices.
     
  4. Jul 19, 2014 #3

    Fredrik

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    There's a simple solution that involves two orthonormal bases, one that's orthonormal with respect to <,>, and one that's orthonormal with respect to [,].
     
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