## Some properties on the closed subsets in Banach spaces

*A**bdelhakim** Maaden,
Abdelkader Stouti*

** Address.**

Unit\'e de Recherche
Math\'ematiques et Applications, Facult\'e des Sciences
et Techniques, B.P.
523, Beni
- Mellal, Maroc** **

** **

**E-mail:
**hmaaden2002@yahoo.fr

** **

**Abstract****.**

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.

**AMSclassification**.
46B20.

**Keywords**. James Theorem, Bishop-Phelps Theorem,
smooth variational principles.