Let $\lbrace f_n \rbrace$ be a sequence of functions in $L^2([a,b])$ for some real closed interval $[a,b]$. In general, if $f_n \rightarrow f$ pointwise, does the sequence also converge in norm?
If the sequence is bounded, it is true by DCT. Also, it is easy to find a counter example in $L^2(\mathbb{R})$. But is it true in general in $L^2([a,b])$? I believe it is not, but I have not been able to build a counter example.
Any help would be appreciated.