Theorems on lifting vector-valued functions

Let $T$ be a set. Let $X$ and $Y$ be locally convex spaces, $L(X,Y)$ the space of linear maps of $X$ into $Y$, and $K\colon T\to L(X,Y)$ some map. A lifting theorem is an assertion that for each $g\colon T\to Y$ from some class of maps there exists a map $f\colon T\to X$, of the same class, such that $K(t)f(t)=g(t)$ for all $t\in T$. In this paper lifting theorems are proved for the classes of continuous, continuously differentiable a finite number of times, and infinitely differential maps.
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