Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int\_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$


or any reasonable Euclidean norm such that bounded functions have a finite norm.


Can we construct a continuous function mapping from $S$ to the space of distributions over $\mathbb{R}$ such that periodic functions of period $\tau$ are mapped to $\delta\_{\tau}$?


The motivation is to define some reasonable notion of the fundamental of a signal.



Fundamental of a signal
Make a sequence of functions of period $2\pi$ whose limit is of period $\pi$. Your map is discontinuous.


