Weak Convergence in Hilbert Spaces

# Weak Convergence in Hilbert Spaces

Recall that if $X$ is a normed linear space then a sequence of points $(x_n)$ in $X$ is said to weakly converge to $x \in X$ if for every $f \in X^*$ we have that:

(1)
\begin{align} \quad \lim_{n \to \infty} f(x_n) = f(x) \end{align}

Recall from The Riesz Representation Theorem for Hilbert Spaces page that if $H$ is a Hilbert space then $f_g : H \to \mathbb{R}$ defined for each $h \in H$ by $f_g(h) = \langle h, g \rangle$ is bounded linear functional on $H$. Furthermore, the Riesz representation theorem for Hilbert spaces says that every bounded linear functional on $H$ is of this form. That is, if $f \in H^*$ then there exists a $g \in H$ such that for every $h \in H$;

(2)
\begin{align} \quad f(h) = \langle h, g \rangle \end{align}

Therefore, we have the following characterization for weak convergence in a Hilbert space.

 Proposition 1: Let $H$ be a Hilbert space. Then a sequence $(x_n)$ in $H$ weakly converges to $x \in H$ if for all $g \in H$, $\displaystyle{\lim_{n \to \infty} \langle x_n, g \rangle = \langle x, g \rangle}$.