A problem on $Z$-valued functions, where $Z$ is a Banach space.

Question

Let $X$, $Y$ Banach spaces, $\mathcal{L}(X,Y)$ the space of linear bounded operators from $X$ to $Y$, if $f:(0,T)\to \mathcal{L}(X,Y)$ is a continuous function and for every $t \in (0,T)$ the function $g:(0,t)\to X$ is also continuous, can we conclude that the function $$ F:(0,t)\times (0,T) \to Y $$ $$ (s,t) \mapsto f(t-s)(g(s)) $$ is also continuous?

Answer 1

I'll assume the conditions I mentioned in the comments otherwise it's not even clear how you can talk about continuity when the domain itself is changing.

Consider the evaluation map $\text{ev}:L(X,Y)\times X\to Y$, $\text{ev}(T,x):=T(x)$. You can easily verify this is a bounded bilinear map, and hence continuous. Your map is just a composition $F(s,t)=\text{ev}\bigg(f(t-s),g(s)\bigg)$, and hence is continuous on the set $U$. To be even more explicit, it is the composition $(s,t)\mapsto (f(t-s),g(s))$ (a mapping into a product space) with the evaluation map.