I`m having a homework question that goes like this: $X$ is a Hilbert space, a complete inner product space, show that $B(X)$ is not a Hilbert space.
My only question for now is what does $B(X)$ mean? My hunch is that that is the space of all bounded linear operators, and if I'm right please give a small example.
@GiuseppeNegro is correct. Remember that a linear operator being bounded is equivalent to it being continuous so you could equivalently view $B(X)$ as the continuous linear operators instead. Furthermore, given two Banach spaces $Y,Z$, it is known that $B(Y,Z)$ is a Banach space. All Hilbert spaces are necessarily Banach spaces so your goal here is to show that while $B(X)$ is a Banach space, it is not a Hilbert space. You can find a hint for the problem in the spoiler text below.