Convergent in Hilbert Spaces

Question

Let $H$ be Hilbert space for fixed $h \in H$. Define $L:H \rightarrow \mathbb{C}$ be bounded linear functional by $L(x):=<x,h>$ , for any $x \in H$. If we have $<x_n , y> \rightarrow <x,y>$. Can we prove that $x_n \rightarrow x$, where $y \neq 0$.

Answer 1

Counterexample- let $y = (1,0)\in\mathbb R^2$ with the usual dot product and let $x_n$ be any sequence with 0 first coordinate.

More generally as long as $\dim H>1$, if $x_n,x \in \operatorname{span}\{y\}^\perp$ then all inner products are 0, no matter if $x_n$ converge or not.