It is often mentioned that the norm convergence in L2 space does not imply pointwise convergence, however I fail to notice why, as the norm in L2 space must be small if two functions are pointwise close to each other. Is thare an example and proof that shows that for L2 spaces the convergence in norm does not imply pointwise convergence?
In other words:
\begin{equation} \left<f(x)-g(x),f(x)-g(x)\right> = \int (f(x) - g(x))^2 \, dx \longrightarrow 0 \end{equation} does not imply \begin{equation} |g(x) - f(x)| = 0 \end{equation}
