function in $L^1\setminus L^2$

Question

I'm looking for an example of a function which belongs to the Banach space $L^1$ (i.e $\int|f|< \infty$) but is not in $L^2$ (so $\int|f|^2$ is unbounded).

Does anyone know such a function?

Answer 1

Try $f(x) = \frac{1}{\sqrt{x}}$ on (0,1)

Answer 2

If the domain is infinite (I'll pick $(0,\infty)$), then you want a function with an integrable singularity like $x^{-\frac{1}{2}}\exp(-x)$.