Question about weak convergence in Hilbert space

[Sardin]

The Question is as follows:

let A be a bounded domain in R^n and
Xm a series of real functions in L^2 (A).
if Xm converge weakly to X in L^2(A)
and (Xm)^2 converge weakly to Y in L^2(A)
then Y=X^2.

i don't know if the above theorem is true and could sure use any help i can get.
if anyone has any proof please post it... thanks.

[jimmysnyder]

As I am not experienced in these matters, this proof should be checked for errors by one of the mentors.

For starters, I will remind everyone what the definition of weak convergence in L^{2}(A) is:

A sequence {X_{n}} is said to converge weakly to X (written X_{n} \stackrel{w}{\rightarrow} X) if for all functions Z in L^{2}(A), we have \int X_{n}Z \rightarrow \int XZ

Next, I will show that if a sequence {X_{n}} in L^{2}(A) converges weakly, then the sequence of integrals |\int X_{n}| is bounded. Just let Z be the constant function Z(x) = 1. Then by the definition of weak convergence,

\int X_{n} = \int X_{n}Z \rightarrow \int XZ = \int X

and \int X_{n} convergent means |\int X_{n}| is bounded.

Next, I will point out that there is a theorem that says that two functions X and Y in L^{2}(A) are equal if for all Z in L^{2}(A) we have

\int XZ = \int YZ

Finally, I get to the proof.

Let M be the bound on |\int X_{n}|. Let L = |\int X|. Let Z be in L^{2}(A), and choose \epsilon > 0.

Since {X_{n}} \stackrel{w}{\rightarrow} X, we can find N_{1} such that for all n \ge N_{1} we have |\int X_{n}Z - \int XZ| < \frac{\epsilon}{3L}.

Also, we can find N_{2} such that for all n \ge N_{2} we have |\int X_{n}Z - \int XZ| < \frac{\epsilon}{3M}.

Since {X_{n}^2} \stackrel {w}{\rightarrow} Y we can find N_{3} such that for all n \ge N_{3} we have |\int X_{n}^{2}Z - \int YZ| < \frac{\epsilon}{3}.

Let N be the maximum of N_{1}, N_{2}, and N_{3}, then for all n > N we have

|\int X^{2}Z - \int YZ|
\le |\int X^{2}Z - \int X_{n}XZ| + |\int X_{n}XZ - \int X_{n}^{2}Z| + |\int X_{n}^{2}Z - \int YZ|
\le |\int X||\int XZ - \int X_{n}Z| + |\int X_{n}||\int XZ - \int X_{n}Z| + |\int X_{n}^{2}Z - \int YZ|
< L\frac{\epsilon}{3L} + M\frac{\epsilon}{3M} + \frac{\epsilon}{3} = \epsilon

So, X^{2} = Y

Q.E.D.