It is shown that under natural assumptions, there exists a linear functional does not have supremum on a closed bounded subset. That is the James Theorem for non-convex bodies. Also, a non-linear version of the Bishop-Phelps Theorem and a geometrical version of the formula of the subdifferential of the sum of two functions are obtained.
We present the full descriptive characterizations of the strong McShane integral (or the variational McShane integral) of a Banach space valued function defined on a non-degenerate closed subinterval of in terms of strong absolute continuity or, equivalently, in terms of McShane variational measure generated by the primitive of , where is the family of all closed non-degenerate subintervals of .
Mean value inequalities are shown for functions which are sub- or super-differentiable at every point.
We show that if U is a balanced open subset of a separable Banach space with the bounded approximation property, then the space ℋ(U) of all holomorphic functions on U, with the Nachbin compact-ported topology, is always bornological.