Eva Matousková, Charles Stegall
The Structure of the Fréchet Derivative in Banach Spaces
Preprint series: ESI preprints

MSC:

46B20 Geometry and structure of normed linear spaces

46G05 Derivatives, See also {58C20, 58C25}

46B22 Radon-Nikodym, Kreuin-Milman and related properties, See also {46G10}

Abstract: Our analysis of Fr\'echet differentiable functions
obtains results of the following type.
An operator is factorizable if it factors through
an Asplund space or satisfies any of the conditions
of \cite{SRNP2}. Suppose that $X$ is a Banach space,
$W$ is an open subset of the Banach space $Z$
and $\beta : W \to Z$ is Fr\'echet differentiable at every point.
The set
$$
\{w \in W : \beta ^\prime (w) \text {
factorizble}\}
$$
is a dense subset of $W$ if and only if for any continuous and
convex function $\phi : X \to \Re$ the composed function
$\phi \beta $ is Fr\'echet
differentiable on a dense G$_\delta $
subset of $W$. We observe that if a continuous function
is Fr\'echet differentiable everywhere on an open set then the
derivative is in the first Baire class.

Keywords: Frechet derivative, Banach spaces