Let $(X, |\cdot|_1)$ and $(Y, |\cdot|_2)$ be Banach spaces. Let $\mathcal L(X, Y)$ be the space of all continuous linear maps from $X$ to $Y$. We endow $\mathcal L(X, Y)$ with the operator norm $\|\cdot\|$. Then $(\mathcal L(X, Y), \|\cdot\|)$ is a Banach space. I would like to ask if the following statement is true, i.e.,
Statement If $X, Y$ are Hilbert spaces, then so is $\mathcal L(X, Y)$.
If $Y =\mathbb R$, the statement is true by Riesz representation theorem. Thank you so much for your elaboration!
Best Answer
The parallelogram-identity fails already for $2\times 2$-matrices: Consider the Hilbert space $X=Y=\mathbb{R}^2$. Note that for symmetric matrices $\|A\|= r(A)$ ($r$ the spectral radius). Set $A=\rm{diag}(2,1)$ and $B=\rm{diag}(1,2)$. Then $$ \|A+B\|^2+\|A-B\|^2=3^2 + 1^2 =10, $$ but $$ 2(\|A\|^2+\|B\|^2)=2(2^2+2^2)=16. $$