# Math Help - Hilbert Space

1. ## Hilbert Space

Let $H$ be a Hilbert space. Show that $H^*$ is also a Hilbert space with respect to the inner product defined by $(f_x,f_y) = (y,x)$.
( $x,y \in H , f_y,f_x \in H^*$ )

I had proven that it is an inner product space and my problem is how to show that $H^*$ is a Hilbert space.

Can anyone help?

2. Frechet-Riesz theorem assures that $H^*$ is a Banach space with the operator norm.

Another thing you could use is that if $X,Y$ are normed spaces and $Y$ is a Banach space then $\mathcal{B} (X,Y) := \{ f:X\rightarrow Y : f \ is \ linear \ and \ continous \}$ is also a Banach space with the operator norm (which in your case coincides with the one induced by the inner product).