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## Homework Statement

Let [itex]H[/itex] be a Hilbert space. Prove [itex]\Vert x \Vert = \sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert}[/itex]

## The Attempt at a Solution

First suppose [itex]x = 0[/itex]. Then we have [itex]\sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert} = \sup_{0\neq y\in H}\frac{\vert (0,y) \vert}{\Vert y \Vert} = \sup_{0\neq y\in H}\frac{\vert 0 \vert}{\Vert y \Vert} = 0 = \Vert 0 \Vert[/itex].

Now suppose [itex]x \neq 0[/itex]. Then [itex]\Vert x \Vert = \sqrt{(x,x)} = \frac{\sqrt{(x,x)} \cdot \sqrt{(x,x)}}{\sqrt{(x,x)}} = \frac{\vert (x,x)\vert}{\Vert x \Vert} \leq \sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert}[/itex].

Now I just can't do the reverse inequality. Any help is much appreciated.