I'm sorry about this question but the more I think about this, the more I feel ignorant.
How can I, by hand, create a sequence of functions $(f_h)_h$, let's say in $L^2((01))$, coverging, under the $\left\lVert \cdot\right\rVert_{L^1}$ norm, to $\frac{1}{\sqrt{x}}$, for example.
This specific example was meant to show that $L^2((0,1))$ is not closed in $L^1((0,1))$, but this question would like to be way general, as I find myself always in trouble in creating explicitely sequences of functions. I'd really like to see the method for which create sequences, rather than simply read a specific answer as it won't help me grow, I think.
Any solution, hint or reference would be much appreciate, thanks in advance.
If $f$ is an integrable function, let $f_h=f\mathbf 1\{\lvert f\rvert \leqslant h\}$; then $f_h\in\mathbb L^2$ and by dominated convergence, $f_h\to f$ in $L^1$.