I am learning about Schwartz space $\mathcal{S}$ on $\mathbb{R}$. I showed that for each $y \in \mathbb{R}$, the operator $\tau_y: \mathcal{S} \rightarrow \mathcal{S}$ given by $\tau_yf(x) = f(x -y)$ is continuous.
For two locally convex spaces $X,Y$ with a countable family of seminorms that induce the topology, define $L(X,Y)$ to be the set of continuous linear maps from $X$ to $Y$. Is there a topology on $L(X,Y)$ that is most common or makes the most sense? Preferably one that makes composition continuous if possible?