Projection On L2 Space

**mathematicalbagpiper** · November 30th 2010, 08:01 AM

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?

**Ackbeet** · November 30th 2010, 08:31 AM

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,

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

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