Hint for problem V.1.8

To see successive hint(s), highlight the item(s) below.

For an analytic proof:

Compute ||h_n - h||².
Now use the fact that h_n converges weakly to h when computing the limit as n goes to infinity.

For a geometric proof:

Assume by contradiction that h_n are at distance at least a from ||h||, but have norm about the same as ||h||. Where are they located?
Use the "uniform convexity" of the balls in a Hilbert space to separate h from h_n by a hyperplane.

Maybe both proofs can be adapted to L^p, p > 1 and finite.
Here is a reference for the geometric proof, and the definition involved.