
Projection On L2 Space
Let $\displaystyle V=L^2(\Omega)$ and for $\displaystyle r>0$:
$\displaystyle K=\{v\in V : v_{L^2(\Omega)}\leq r\}$.
For any $\displaystyle u\in V$, find its projection on $\displaystyle K$.
I have no idea where to start. I know that the projection map has certain properties (linear, selfadjoint, operator norm of 1, whatever), but is there any specific formula to find it?

Hmm. Interesting problem. The trick is that applying the projection operator twice should be the same as applying it just once $\displaystyle (P^{2}=P)$. Couldn't you multiply each $\displaystyle u$ by the scalar
$\displaystyle C_{u}=\dfrac{r}{\u\_{L^{2}(\Omega)}}?$
So,
$\displaystyle Pu=\dfrac{ur}{\u\_{L^{2}(\Omega)}}.$