norms of operator

Question

I am stuck on a question on the operator norm. If $L$ is a bounded linear operator $L:H\rightarrow H$ where $H$ is a Hilbert space. How would you show that $$\|L\|\leq \sup_{\|u\|=\|v\|=1}(Lu,v)$$ any hints? Thanks,

Answer 1

Hint : $$\sup_{\|u\|=\|v\|=1} (Lu,v) \geq \sup_{\|u\|=1} (Lu, \frac{Lu}{\|Lu\|})$$

Answer 2

Hint: if $A \subseteq B \subseteq \Bbb R$, $A$ is non-empty, and $B$ is bounded above, then $$ \sup(A) \leq \sup(B) $$