I'm trying to solve this question:
Let $Y$ be the subspace of $C[-1,1]$ consisting of even functions. Prove that $Y$ is closed and that for any $f\in C[-1,1]$ the function $Pf(x)=\frac{f(x)+f(-x)}{2}$ is the unique element of $Y$ closest to $f$, i.e $$\text{inf}_{g\in Y}\lVert f-g \rVert=\lVert f- Pf \rVert$$
The excercise doesnt say with which norm but I think one should assume that it's the infinity norm? I know how to show that it is closed, but I don't know how to tackle the minimization problem. If I'm not mistaken, $C[-1,1]$ isn't unifomly convex with the supremum norm and there's no guarantee for a unique element satisfying the infimum. I'm thinking that maybe one can use polynomial approximation:
If $\mathbb{P_n}$ is the space of polynomials of degree n, project $f$ onto $Y \cap \mathbb{P}_n$, and then use some kind of density argument.
I would be very grateful if somebody could help me with this!