I'm working through the details of verifying various facts about the Riesz Isomorphism between a normed vector space $(E, || \cdot ||_E)$ (with an inner product) and its dual E'. One of the things that I want to show is that the map $f_y(x) = (x | y)$ is bounded and, in fact, ||f_y||_{E'} \leq ||y||_E where ||\cdot||_{E'} denotes the operator norm on E'. I believe that I have this worked out but would appreciate a second opinion as I always feel inept when attempting proofs that involve estimation!
First, I claim $f_y$ is bounded (by $||y||_E$) because
$ |f_y(x)| = |(x|y)| \leq ||y||_E \cdot ||x||_E$
where the last inequality follows from Cauchy-Schwarz. Therefore, the operator norm can be meaningfully applied to $f_y$. Furthermore,
||f_y||_{E'}
$= \sup\{|f_y(x)| : ||x||_E \leq 1 \}$
$= \sup\{|(x|y)| : ||x||_E \leq 1 \}$
$ \leq \sup\{ ||x||_E \cdot ||y||_E \ : ||x||_E \leq 1 \} $
$ =||y|| \sup\{||x||_E : ||x||_E \leq 1 \} $
$ = ||y||_E $
Here, the fourth line follows again from Cauchy-Schwarz.
So, my question is, does this proof work or are there gaps in my reasoning?