# Induced inner product on tensor powers.

Let $$V$$ be a real or complex inner product space with inner product $$\left\langle \cdotp,\cdotp\right\rangle$$. For $$\otimes ^kV$$ define $$\left\langle \cdotp,\cdotp\right\rangle_k$$ by $$\left\langle u_1\otimes ...\otimes u_k,v_1\otimes ...\otimes v_k\right\rangle_k = \left\langle u_1,v_1\right\rangle...\left\langle u_k,v_k\right\rangle$$ on pure tensors, and extend bilinearly.

Is the resulting object $$\left\langle \cdotp,\cdotp\right\rangle_k$$ an inner product on $$\otimes ^kV$$? It is clearly a symmetric bilinear form. Is it non-degenerate and positive definite though?

• do $v_i=u_i$ for all the $i$ and infer – janmarqz Jan 26 at 17:35
• That only gives the result for tensors of the particular form above. – Joshua Tilley Jan 29 at 19:25
• you need to do that in order to check your last question – janmarqz Jan 30 at 3:23