spectral norm of PSD matrices and inner product

Question

Is the operator norm on the space of $n \times n$ real PSD matrices derived from an inner product? In particular, spectral norm for matrix $A \in \mathbb{R}^{n \times n}$ defined as \begin{equation} \| A \| = \max_{x\in \mathbb{R}^{n}: \| x \|\leq 1} \| Ax \|_2, \end{equation} where $ \| . \|_2$ denote the $\ell_2$ norm in $\mathbb{R}^{n}$.

Answer 1

Based on the comment by @user1551, I answer this question.

For the sake of contradiction assume that there exists an inner product $\mathbb{R}^{n\times n}\times \mathbb{R}^{n\times n} \rightarrow \mathbb{R}$ over the vector space of all symmetric matrices in $\mathbb{R}^{n\times n}$. Then, the parallelogram law must hold. Therefore, if an inner product exists, we should be able to show that for every PSD $A$ and $B$, we have \begin{equation} 2 \|A\|^2 + 2 \|B\|^2 = \|A+B\|^2 + \|A-B\|^2. \end{equation}

Lets consider $A = \text{diag}([1,0])$ and $B = \text{diag}([0,1])$. Then, $ \|A\|=1$ , $ \|B\|=1$, $\| A+B\|=1$, and $\| A-B\|=1$. This example shows that parallelogram does not hold. Contradiction.