Gelfand triples/Rigged Hilbert space notation Usually the rigged Hilbert space is denoted by $\mathcal{S} \subset L^2 \subset \mathcal{S}'$, where $L^2$ is a Hilbert space (square integrable functions), $\mathcal{S}$ is the Schwartz space and $\mathcal{S}'$ the space of tempered distributions. It's clear to me that $\mathcal{S}$ is a subset of $L^2$, but why do we write $L^2 \subset \mathcal{S}'$? These two sets have different elements. How is one subset of the other? One has functions and the other has functionals.

This is an abuse of notation. It is true that, as you've stated, $L^2$ is not a subset of $\mathcal S'$. What the notation $L^2\subset \mathcal S$ means is that $L^2$ embeds in $\mathcal S'$