$L^{2}$ convergence, bounded function.

Question

Let $X$ be a metric space and $\mathcal{B}(X)$ be a Borel $\sigma$-algebra on $X$ and $\mu$ be a finite measure on $X$.

We consider continuous functions (denoted by $\{f_{n}\}$) on $X$.

If $f_{n}\to g$ in $L^{2}(X\,;\mu)$ and $g$ is bounded on $X$, then $\{f_{n} \}$ is uniformly bounded or that of similar is hold?

Answer 1

False. Take $f_n$ zero except in a very narrow (area$^2 < 1/n$) and tall (height $>n$) spike. The sequence $f_n\to 0$ in $L^2$ but is unbounded.