Inner product on the k-tensor space

Question

This is homework so no answers please.

The problem is

"Given inner product vector space V, define an inner product on $T^{k}(V)$ by declaring the standard basis $\{e^{*}_{i_{1}}\otimes...\otimes e^{*}_{i_{k}}\}$ to be orthonormal.

Show that this inner product is basis-independent."

Firstly , I am trying to define an inner product on $T^{k}(V)$. Any hints?

Maybe $\left \langle T,S \right \rangle:=sup_{V\otimes V}T\otimes S (v\otimes w)$.

Answer 1

Hint:

If each of the $e^{*}_{i_{1}}\otimes...\otimes e^{*}_{i_{k}}$ is orthogonal to each other and normalized, we should have

$\langle e^{*}_{i_{1}}\otimes...\otimes e^{*}_{i_{k}}, e^{*}_{j_{1}}\otimes...\otimes e^{*}_{j_{k}} \rangle = \delta_{i_1j_1} \cdots \delta_{i_kj_k}$

Now, by ...., you can extend this definition to ....

Answer 2

Have you thought, once the inner product is defined, to rewrite it in more universal terms?

Work first with $V\otimes V$, and see if you can do it.