What does it mean to be absolutely integrable on $\mathbb{R}$ and what are the steps to show that something is absolutely integrable?

Question

I just have a quick question. What does it mean to be absolutely integrable on $\mathbb{R}$ and what are the steps to show that something is absolutely integrable? For example what if we wanted to show that the vector space of functions $f$ in a Schwartz space is absolutely integrable?

Answer 1

$f$ is absolutely integrable on $\mathbb R$ if $$\int_{\mathbb R} |f(x)|\,dx<\infty.$$

For the second question, I'd start by letting $f\in\mathcal{S}(\mathbb R)$ and consider $\int_\mathbb R |f(x)|\,dx$, using what you know about $\mathcal{S}(\mathbb R)$ to show the integral is finite.