I think to have proved the following:
Given $K$ a convex closed(maybe also limited is needed)subset(also curve not just subspaces) of an Hilbert space $H$, is well defined the projection operator $p_K:H \to K$. This operator is continuos for the norm topology. If i didn't miss something, my guess (is not a very formal proof) shows that is sequentially continuous for the weak topology. Is that true?
My guess goes more or less like this: i write my convex set as intersection of half spaces, and i reduce in some way to those.
Before to try to work it out more clearly i'd like to know if anybody knows if the result is actually true. Thanks
Here is a counter-example regarding continuity of the projection in the weak topology. Is is taken from Zarantonello's article I mentioned in the comments.
Let $H=l^2$, $K$ be the closed unit ball. Let $e_k=(0,\dots, 0,1,0,\dots)$ be the $k$-the unit element. Set $$ x_n := e_1 + e_n. $$ Then $x_n\rightharpoonup e_1$. The projection of $x_n$ satisfies $$ p_K(x_n) = \frac1{\sqrt 2}(e_1 + e_n) \rightharpoonup \frac1{\sqrt 2}e_1. $$ Thus the projection of the weak limit of $x_n$ is $e_1$, and it is different from the weak limit of the projections, which is $\frac1{\sqrt 2}e_1$. And $p_K$ is not continuous with respect to the weak topology.