How to solve the following problem:Let f n , f ∈ L 2 ( R...
How to solve the following problem:
Let for all be such that as . Suppose, moreover, that
for all . Then converges to in -norm.
Answer & Explanation
You want to show
Equivalently, you can show
Since we have , we have and since we have we have so that
and hence of course also
Since is a Hilbert space, you can use the parallelogram identity. More generally, you can also use a property of any uniformly convex Banach space. A very nice proof appears in Brezis' book on functional analysis.