Let $X$ and $Y$ be locally convex, Hausdorff topological vector spaces and let $[a,b] \subset \mathbb{R}$. Let $f: [a,b] \to \hom(X,Y)$ be continuous, where $\hom(X,Y)$ is the space of continuous linear maps from $X$ to $Y$ with the topology of uniform convergence on bounded subsets of $X$. Let $H \subset \hom(X,Y)$ be the image of $f$.

Is $H$ always an equicontinuous subset of $\hom(X,Y)?$ This is true in the case where $X$ is barelled by the Banach-Steinhaus theorem, but I do not want to make any additional assumptions on $X$ or $Y$.

If $H$ need not be equicontinuous in general, can we make any modifications to $f$ to ensure it is? e.g. assuming $f$ is infinitely differentiable, or continuous with respect to a different topology on $\hom(X,Y)$. Again, I do not want to make any additional assumptions about the spaces $X$ and $Y$.