I was woundering if there is an easy way to calculate the projection of a Hilbert space into a closed subspace.
Obviously one could write $P:H->C$ that is given by $P(x)=d$ s.a $d=inf||x-v||$ for any $v\in C$. But i'm looking for a way to write the explicit form of $P$, for specific examples. What method should I take in order to find an explicit form?
For example, take $H=L^2(R)$ and $C=\{f\in L^2(R)|f-is-even-a.e\}$. (intuitivly I'd guess that the projection of any $f$ would be a a symmetric function which is identical to $f$ on the half space of R (the positive or the negetive) in which $f$ has larger norm. but then again it's a guess and I don't know how to prove such theories).
Honestly I'd like some kind of method to do such questions.
Thanks so much
The following theorem can be helpful:
Let $H$ be a Hilbert space and $C \subseteq H$ be a closed subspace of $H$.
Then the projection operator $P: H \rightarrow C$ is linear and $y \in C$ satisfies $(y - x, z) = 0$ for any $z \in C$ if and only if $y = Px$.
As for the example, for $f \in H$, let $g \in C$ be defined by $g(x) = \frac{f(x) + f(-x)}{2}$.
Then $(f - g)(x) = \frac{f(x) - f(-x)}{2}$ is odd and $(f - g, h) = 0$ for any $h \in C$.
It remains to show $C$ is a closed subspace.