Question about linear mappings and inner product spaces

[34567]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[jbunniii]

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.

 

[Fredrik]

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 [,].