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I'm learning about Hilbert spaces and found that, in the case of matrices as elements of the space, the Frobenius inner product induce the Frobenius norm.

I'm interested in knowing more examples of matrix norms that satisfy the parallelogram law and thus are induced by an inner product.

I would also appreciate if you could point me to some reference to look at.

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Unless you are not interested in multiplication of matrices you may regard them as elements of $n^2$ dimensional vector space, so there is nothing new here compared with usual finite dimensional spaces. For them there is a complete description of all possible inner products: $$ \langle x, y\rangle =\sum_{i,j=1}^k G_{ij}x_i\overline{y_j} $$ for all $x,y\in \mathbb{C}^k$ and some positive definnite matrix $G$

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