I'm looking for simple, easy examples for the following two phenomena where there is no need to consider the convergence of any subsequences. I'm considering sequences $\{f_n\}_{n=1}^\infty\subset L_2([-\pi,\pi],m)$ where $m$ is the Lebesgue measure.
Pointwise convergence does not imply $L_2$ convergence.
$L_2$ convergence does not imply pointwise convergence.
I see that any example for 2. would also be an example highlighting that $L_2$ convergence does not imply uniform convergence. I already have a proof for the fact that uniform convergence implies $L_2$ convergence, and whilst I have some examples for showing that pointwise convergence does not imply uniform convergence I am finding it hard to adapt them for these two cases here.
There is a standard example for 2). Divide $[-\pi,\pi]$ into two equal sub-intervals $I_{1,1}$ and $I_{1,2}$; the divide these again into equal sub-intervals $I_{2,1},I_{2,2},I_{2,3},I_{2,4}$, and so on. If you arrange the characteristic functions of all these intervals in a single sequence you get an example for 2). For 1) you can use the example given in above comment.