Separating polynomials, uniform analytical and separating functions

We present basic results of the theory of separating polynomials, uniformly analytic and separating functions on separable real Banach spaces. We consider basic properties of separating polynomials, uniformly analytic and separating functions. We indicate a relation between weak polynomial topology and norm topology of a space, provided it admits a separating polynomial. We present sufficient conditions for the existence of analytic and uniformly separating functions. We investigate a composition of an uniformly analytic and separating function and a linear mapping.