If I have a function $f$ belongs to the Schwartz space, i.e. $f\in \mathcal{S}$, how can I prove $f\in L^p$ ?
I know that $\mathcal{S}\subset L^p$ hence the above should make sense. But I need a proof.
If I just take a function from $\mathcal{S}$, then compute its $L^p$ norm, and I can see clearly it is bounded hence $f\in L^p$, I mean $$ \int|e^{-a|x|^2}|^p<\infty$$ is obvious. But is this a proof?
A function in $\mathcal{S}$ decays faster than the reciprocal of any polynomial at infinity and is bounded. (This is usually built into the definition.) So its magnitude is bounded by some $M_1$ on the ball of some radius $R$ centered at the origin, and by $M_2/|x|^N$ outside this ball. Choose $N$ sufficiently large (depending on the dimension and $p$) to get your result.