# Question about weak convergence in Hilbert space

1. Mar 18, 2005

### 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.

2. Mar 19, 2005

### Jimmy Snyder

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.

Last edited: Mar 19, 2005