Originally Posted by

**surjective** Hey,

For any $\displaystyle \bf{v} \in V$ i would like to show the following:

$\displaystyle

\Vert \bf{v} \Vert = \sup \{ | \langle \bf{v},\bf{w} \rangle| \big| \bf{w}\in V, \Vert w\Vert=1\}

$

I have done the following: To show an equality one can show that the case is true for both $\displaystyle \geq$ and $\displaystyle \leq$. For the $\displaystyle \geq$-case I have used the Cauchy-Schwarz inequality:

$\displaystyle |\langle \bf{v},\bf{w} \rangle| \leq \Vert \bf{v} \Vert \Vert \bf{w} \Vert$

Now since $\displaystyle \Vert \bf{w}\Vert=1$ we have that:

$\displaystyle |\langle \bf{v},\bf{w} \rangle| \leq \Vert \bf{v} \Vert$

Taking the $\displaystyle \sup$ on both sides of the inequality yields:

$\displaystyle \sup|\langle \bf{v},\bf{w} \rangle| \leq \sup\Vert \bf{v} \Vert$ That should say $\displaystyle \color{red}\sup|\langle \bf{v},\bf{w} \rangle| \leq \| \bf{v} \|$ – see HallsofIvy's comment.