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.