Hello! Just found this help forum and I'm overjoyed. I'm studying functional analysis all by myself using a book (Peter D. Lax) and I'll be thrilled over any help given. I've gotten quite far and am now trying to solve some exercises. This is the first one I've had a problem with.

I need to find a seq. of cont. functions g_n:[0,1]->R (R being the set of real numbers) such that:

(1) {g_k} converges weakly to 0 in L^p([0,1]) for all 0 < p < infinity
(2) {g_k} has no weakly convergent subseq in C([0,1]).

(Btw any special tricks on how to approach future problems of this kind? To me it looks odd, I mean... there are a lot of seq., how do I aim at the right one?)

Many Thanks!