Explanation of notation $f(t)\in L_{\infty}$ in a control theory textbook

Question

In a control theory textbook I saw the following notation : $$f(t)\in L_{\infty}$$ Since I am not familiar with this kind of notation could someone explain What does it mean?

Answer 1

Often, it means that $f(t)$ is a bounded function - that there is a definite constant $M$ for which $-M<f(t)<M$ for all $t$.

The notation is for the following reason: Functions have various 'norms', similar to 'distances' between points. Consider $$\left[\int_{-\infty}^{\infty}|f(t)|^pdt\right]^{1/p}$$ for any value $p$, and ask whether this is finite. If it is finite for $p=1$, we call $f(t)$ absolutely integrable, or $L_1$; if $p=2$ we call it square-integrable or $L_2$. It turns out that, in the limit as $p\to\infty$, this expression approaches the maximum value of $f(t)$, so it is finite - and $f(t)$ is $L_{\infty}$ - if $f(t)$ is bounded.