Riesz representation theorem: show isometry

Question

Let $X$ be a Hilbert space and $J:X\rightarrow X',J(X):=(\cdot,x)$ where $X'$ is the dual space of $X$.

I have to show that $\|J(x)\|_{\sup}=\|x\|$.

''$\leq$'' is clear by the Cauchy-Schwarz inequality

For ''$\geq$'' I somehow have to use $|J(x)(x)|=|(x,x)|=\|x\|^{2}$ but I don't see how.

Answer 1

For $x \neq 0$, $|J(x) (\frac x {\|x\|})|=\|x\|$ so $\|J(x)\| \geq \|x\|$.