Let $X$ be a Tychonoff space. It is well known, that for a family of scalar functions equicontinuity + pointwise boundedness imply relative compactness in $C(X)$ (with compact-open topology). It is also well known, that the converse holds, if we assume that $X$ is compactly generated. It is natural that we need some assumptions on $X$, since properties of compact-open topology on $C(X)$ depend on the status of compact sets in $X$ (although I don't know any counterexamples). I am interested which assumption about $X$ is sharp. More precisely: if any relatively compact in $C(X)$ family of functions is equicontinuous, what can be said about $X$?
Another question, much more down-to-earth, also involving linear structure of $C(X)$.
Let $E$ be an absolutely convex (that is $aE+bE\subset E$, for any $|a|+|b|\le 1$) relatively compact subset of $C(X)$ and let $f_E$ be a function on $X$ defined by $f_E(x)=\sup_{f\in E} f(x)$. Is it true, that $f_E$ is continuous on $X$?