The Radon-Riesz Theorem

Theorem (The Radon-Riesz Theorem): Let $H$ be a Hilbert space. Then a sequence $(h_n)$ in $H$ norm converges to $h \in H$ if and only if $(h_n)$ weakly converges to $h$ and $\displaystyle{\lim_{n \to \infty} \| h_n \| = \| h \|}$.

• Proof: $\Rightarrow$ Suppose that $(h_n)$ norm converges to $h$. Since the norm convergence implies weak convergence we have that $(h_n)$ weakly converges to $h$. Furthermore, since $(h_n)$ norm converges to $h$ we have that $\displaystyle{\lim_{n \to \infty} \| h_n - h \| = 0}$. Since $|\| h_n \| - \| h \|| \leq \| h_n - h \|$ we conclude that $\displaystyle{\lim_{n \to \infty} \| h_n \| = \| h \|}$ as well.

• $\Leftarrow$ Suppose that $(h_n)$ weakly converges to $h$ and $\displaystyle{\lim_{n \to \infty} \| h_n \| = \| h \|}$. Observe that:

(1)

\begin{align} \quad \| h_n - h \|^2 &= \langle h_n - h, h_n - h \rangle \\ &= \langle h_n, h_n \rangle - 2\langle h_n, h \rangle + \langle h, h \rangle \\ &= \| h_n \|^2 - 2 \langle h_n, h \rangle + \| h \|^2 \end{align}

• Since $(h_n)$ weakly converges to $h$, for all $f \in H^*$ we have that $\displaystyle{\lim_{n \to \infty} f(h_n) = f(h)}$. But since $H$ is a Hilbert space, by The Riesz Representation for Hilbert Spaces, we have that $\displaystyle{\lim_{n \to \infty} \langle h_n, h \rangle = \langle h, h \rangle = \| h \|^2}$. Hence:

(2)

\begin{align} \quad \lim_{n \to \infty} \| h_n - h \|^2 = \lim_{n \to \infty} [ \|h_n \|^2 + \| h \|^2] - 2 \lim_{n \to \infty} \langle h_n, h \rangle = 2 \|h\|^2 - 2\|h^2 \| = 0 \end{align}

• Therefore $(h_n)$ norm converges to $h$. $\blacksquare$