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?

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

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