# Reference request: Bounded operators are not a Hilbert space

I believe that the following is true: Let $$X$$ and $$Y$$ be normed spaces, both of dimension at least $$2$$. Then, the space of bounded linear operators $$L(X,Y)$$ is not a Hilbert space. Is there a nice reference for this result?

Sketch of a proof: Suppose that $$L(X,Y)$$ is a Hilbert space. By considering $$y \otimes x^* \in L(X,Y)$$, $$(y \otimes x^*)(x) := \langle x^*, x\rangle_X y$$ for fixed $$y \in Y$$ or fixed $$x^* \in X^*$$, $$X^*$$ and $$Y$$ are subspaces of $$L(X,Y)$$ and, thus, pre-Hilbert spaces. Thus, also $$X$$ is pre-Hilbert. By considering orthonormal sets $$\{e_1, e_2\} \subset X$$ and $$\{f_1, f_2\} \subset Y$$ it is easy to check that the parallelogram identity in $$L(X,Y)$$ fails and this yields a contradiction.

• Perhaps this helps math.stackexchange.com/questions/4451818/… Commented May 19, 2022 at 14:17
• Your proof is correct. In the linked question it is assumed that $X,Y$ are Hilbert, which is not the case here.
– daw
Commented May 20, 2022 at 5:52