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?