Let $A$ be a Banach space and $A_0\subseteq A$ a dense subset (linear subspace if necessary). Write $$F_A=\{\text{continuous functions } f\colon[0,1]\to A\},$$ $$F_{A_0}=\{\text{continuous functions } f\colon[0,1]\to A_0\}.$$ Equip $F_A$ with the uniform norm, i.e. $$\|f\|=\sup_{t\in[0,1]}\|f\|_A.$$ Question 1: Is $F_{A_0}$ dense inside $F_A$?
Now let's consider a more general situation, where $X$ and $Y$ are topological spaces. Let $Y_0\subseteq Y$ be a dense subset, and define $$F_Y=\{\text{continuous functions } f\colon X\to Y\},$$ $$F_{Y_0}=\{\text{continuous functions } f\colon X\to Y_0\}.$$
Equip $F_Y$ with the compact-open topology.
Question 2: Under what conditions on $X$ and $Y$ would $F_{Y_0}$ be dense inside $F_Y$? (We might require $X$ to be compact, Hausdorff, etc.)
Thoughts: It seems fairly natural to think that $F_{A_0}$ would be dense in $F_A$, since for each $f\in F_A$ and each $t\in[0,1]$, $f(t)$ can be approximated by something in $A_0$. But it's not clear to me whether this can be extended to a continuous approximation, say $f_0\in F_{A_0}$, of $f$, since the $f(t)$ are not a priori continuous in $t$. Perhaps there is better way to think about the problem, or maybe a way to select the $f(t)$ more carefully?
Added later: Considering the comments, let's impose the condition that $A_0$ is a linear subspace. Then Q1 asks whether a continuous section of a Banach bundle over $[0,1]$ can be approximated uniformly by continuous sections of time kind of ``subbundle".