If $f \neq 0$ is a continuous functional linear on Hilbert space, is it possible that $f(v_n) \to 0$ and $\|v_n\| \to \infty$

Question

Let $H$ be a infinite dimensional Hilbert space over the complex plane $\mathbb{C}$

Let $f: H \to \mathbb{C}$ be a continuous linear functional on $H$ such that $f \neq 0$

Let $\{v_n\}_{n \in \mathbb{N}} \subset H$ be a sequence in $H$ such that $\|v_n\| \to \infty$

My question: Is it possible that $f(v_n) \to 0$ ?

Thanks.

Answer 1

It is possible. Choose $x\in\ker f$, $y\notin\ker f$ normalized with $x\perp y$. Let $z_n= nx+\frac1ny$. Then $\|z_n\|\to\infty$ and $f(z_n)=\frac1nf(y)\to0$.