Let be a Hilbert space. Show that is also a Hilbert space with respect to the inner product defined by .
( )
I had proven that it is an inner product space and my problem is how to show that is a Hilbert space.
Can anyone help?
Frechet-Riesz theorem assures that is a Banach space with the operator norm.
Another thing you could use is that if are normed spaces and is a Banach space then is also a Banach space with the operator norm (which in your case coincides with the one induced by the inner product).