# Let $X, Y$ be Hilbert spaces. Is $\mathcal L(X, Y)$ togerther with the operator norm a Hilbert space?

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!

## 2 Answers

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 $$2(\|A\|^2+\|B\|^2)=2(2^2+2^2)=16.$$

Assume the operator norm is associated with an inner product. For a nontrivial projection $$P$$ we have $$1=\|(I-P)\pm P\|^2 \\ =\|I-P\|^2 +\|P\|^2\pm 2{\rm Re}\,\langle I-P,P\rangle\\ =2 \pm 2{\rm Re}\,\langle I-P,P\rangle$$ Hence $$2{\rm Re}\,\langle I-P,P\rangle=\pm 1$$ a contradiction.