Why do square integrable functions have to be isolated at infinity?

Question

Why the functions of square integrable vanish at infinity? I have read about quantum mechanics, is a condition for statistical interpretation and these functions live in Hilbert space, but I have been searching because the integrable square functions vanish at infinity and I have not found anything, is there a demonstration or where it gives me a reason for this? Has it to do with measurement theory?

Answer 1

They don't; if $f$ is supported on $[0,1]$ and has squared integral $1$ then $g(x)=\sum_{k=1}^\infty f(k^2(x-k))$ is square integrable and doesn't vanish at infinity. (The summands here are the same shape as $f$ but shifted $k$ steps to the right and then squished horizontally by a factor of $k^2$ without being squished vertically.)

These don't come up in physics but it is for physical reasons, not on the grounds of square integrability per se.