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 $f$.


1. 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 $Y$.
2. 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 $Y$.



Equicontinuity of continuous families of maps between topological vector spaces
Assume that there is a sequence $T\_n\in hom(X,Y)$ which converges to $0$ uniformly on all
bounded subsets of $X$ but is not equicontinuous (such a sequence should exist if $X$ fails to be $c\_0$-*quasibarrelled*, see chapter 8.2 of the book *Barrelled Locally Convex Spaces* of J. Bonet and P. Perez Carreras). I believe that the function $f:[0,1] \to hom(X,Y)$
defined by $f(0)=0$, $f(1/n)=T\_n$ and affine-linear interpolation (e.g. $f(t/n + (1-t)/(n+1))
= tT\_n + (1-t) T\_{n+1}$ for $t\in [0,1]$ and $n\in \mathbb N$) should be continuous
but the range is not equicontinuous. It should also be possible to make a smooth variant out of this example.


