# Projection On L2 Space

• Nov 30th 2010, 09:01 AM
mathematicalbagpiper
Projection On L2 Space
Let $V=L^2(\Omega)$ and for $r>0$:

$K=\{v\in V : ||v||_{L^2(\Omega)}\leq r\}$.

For any $u\in V$, find its projection on $K$.

I have no idea where to start. I know that the projection map has certain properties (linear, self-adjoint, operator norm of 1, whatever), but is there any specific formula to find it?
• Nov 30th 2010, 09:31 AM
Ackbeet
Hmm. Interesting problem. The trick is that applying the projection operator twice should be the same as applying it just once $(P^{2}=P)$. Couldn't you multiply each $u$ by the scalar

$C_{u}=\dfrac{r}{\|u\|_{L^{2}(\Omega)}}?$

So,

$Pu=\dfrac{ur}{\|u\|_{L^{2}(\Omega)}}.$