Reference for Weak convergence in hilbert space

Question

I want to understand weak convergence in hilbert space. Can somebody give me a reference for that.

Thanks in advance.

Answer 1

I took an undergraduate functional analysis course which used the text "Introductory Functional Analysis with Applications" by Erwin Kreyszig. Weak convergence is discussed in Chapter 4. I think it is easy to follow and suitable for self-learners.

Let me also talk a little bit concerning your question. In a Hilbert space $H$, every bounded linear functional $T \in H^*$ can be characterized by an element $y \in H$, in the sense that

$$Tx = (x, y) \text{ for } x \in H $$

by F. Riesz Representation Theorem on a Hilbert space. A sequence $(x_n) \subseteq H$ is said to converge to $x \in H$ weakly if $Tx_n \rightarrow T_x \text{ for } T \in H^*$. As a consequence of the F. Riesz Representation Theorem, $(x_n) \subseteq H$ converge to $x \in H$ weakly if and only if

$$ (x_n, y) \rightarrow (x,y) \text{ for } y \in H$$

Strong convergence implies weak convergence (easy to check) but the reverse is not true. Consider $H = L^2(-\pi,\pi)$ equipped with the $L^2$ norm and $f_n(x) = sin(nx) \text{ for } x \in [-\pi, \pi]$. Then $(f_n) \subseteq H$ converge to $0$ weakly but not strongly.