How to solve the following problem:Let f n , f ∈ L 2 ( R...

veneciasp

veneciasp

Answered

2022-07-07

How to solve the following problem:

Let f n , f L 2 ( R d ) for all n 1 be such that f n 2 f 2 as n . Suppose, moreover, that
f n g f g
for all g L 2 ( R d ). Then f n converges to f in L 2 -norm.

Answer & Explanation

Oliver Shepherd

Oliver Shepherd

Expert

2022-07-08Added 24 answers

You want to show
f f n 2 0
Equivalently, you can show
( f f n ) 2 d μ = f f n 2 2 0
We have
f f n 2 2 = ( f f n ) 2 d μ = f 2 d μ 2 f f n d μ + f n 2 d μ
Since we have f n 2 f 2 , we have f n 2 d μ f 2 d μ and since we have f n g d μ f g d μ we have f f n d μ f 2 d μ so that
f 2 d μ 2 f f n d μ + f n 2 d μ f 2 d μ 2 f 2 d μ + f 2 d μ = 0
that is,
f f n 2 2 0
and hence of course also
f f n 2 0
bandikizaui

bandikizaui

Expert

2022-07-09Added 7 answers

Since L 2 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.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?