Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert space, and let $(u_n), (v_n)$ be two weakly convergent sequences in $H$ with limits $u$ and $v$ respectively, i.e. $\langle u_n, y \rangle \to \langle u, y \rangle, \langle v_n, y \rangle \to \langle v, y \rangle \quad \forall y \in H$ for $n \to \infty$. I now want to show that $\langle u_n, v_n \rangle \overset{n \to \infty}\longrightarrow \langle u, v \rangle$.
Now my initial thought was to add in a zero and use the inner product properties, i.e.
$$ |\langle u, v \rangle | – \langle u_n, v_n \rangle | = | \langle u, v \rangle – \langle u, v_n \rangle + \langle u, v_n \rangle – \langle u_n, v_n \rangle | \\
\leq |\langle u, v – v_n \rangle | + | \langle u – u_n, v_n \rangle| \\$$
Now the left term in this equation should converge to $0$ for $n \to \infty$ because of the weak convergence of $v_n$; with the right term however, I'm not sure what to do since the $n$ still appears on both sides. I suppose I now want to show that $|\langle u – u_n, v_n \rangle| \overset{n \to \infty}\longrightarrow 0$ aswell to complete the proof, but I'm not sure how to do that. Since we're in a Hilbert space here, I thought about using the Riesz representation theorem somehow, but I'm not sure how it would help me.
Best Answer
This is false. If $\{e_n\}$ is an orthonormal basis then $e_n\to 0$ weakly but $\langle e_n, e_n \rangle$ does not tend to $0$.