Suppose we have (possibly infinite-dimensional) vector spaces $A, B,$ and $C$. For linear maps $f\colon A\to B$ and $g\colon A\to C$ such that $\ker(g)\subset \ker(f)$, we have $f=h\circ g$ for a linear map $h\colon C\to B$, which follows from quotienting $A$ by $\ker(g)$.
As a (regretful) non-algebraist, I would not know where or how to search for analytic properties of $h$ if the vector spaces are normed (/topologic) and $f$ and $g$ are bounded (/continuous).
I would be grateful for either supply of a keyword or a good reference.