Compact factorization of differentiable mappings

Results on factorization (through linear operators) of polynomials and holomorphic mappings between Banach spaces have been obtained in recent years by several authors. In the present paper, we obtain a factorization result for differentiable mappings through compact operators. Namely, we prove that a mapping $f:X\to Y$ between real Banach spaces is differentiable and its derivative $f'$ is a compact mapping with values in the space ${\mathcal K}(X,Y)$ of compact operators from $X$ into $Y$ if and only if $f$ may be written in the form $f=g\circ S$, where the intermediate space is normed, $S$ is a precompact operator, and $g$ is a Gâteaux differentiable mapping with some additional properties. We also show that if $f'$ is uniformly continuous on bounded sets and takes values in ${\mathcal K}(X,Y)$, then $f'$ is compact if and only if $f$ is weakly uniformly continuous on bounded sets.