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Robert Deville :
Smooth functions on Banach spaces

Natural situations, like the resolution of Hamilton-Jacobi equations,
involve non-smooth functions, and it is therefore desirable to
obtain non-smooth calculus rules.

The existence of enough smooth functions on a Banach space X
is necessary for developping non smooth calculus in X, since it is
a key hypothesis in perturbed minimization principles.
It is also necessary for approximation non smooth functions by smooth functions.

We present in these lectures various techniques of construction
of smooth functions: one can approximate convex functions by
smooth convex functions using the concept of non linear boundaries.
This concept is related to a new minimax inequality which extends
Simons' inequality. We also present the structure of the set of
derivatives of a smooth real valued bump function defined on a Banach space.
This set can coincide with the dual space, even in the non reflexive case.
On the other hand, it can also be very small: P. Hajek proved that if
f is a C^1-smooth function on c_0 with locally uniformly continuous
derivative, then f' is locally compact.

Hajek's theorem can be proved using the notion of strongly
sequentially continuous functions that we shall also present.