@Matt well, it depends on what you want to do... Pointwise (whatever that should mean) is certainly far too weak for most purposes, but e.g. for proving density of smooth functions in $L^p(\mathbb{R}^n)$ you can do with far less. Convergence in the sense of distributions is the "right" way to phrase it since it gives all that you need and can hope for in most contexts. A decent introduction is given in Felder's
notes on MMP chapter 4, but