Abstract
Publisher Summary This chapter discusses the approximation of holomorphic and differentiable functions defined on Banach spaces. In the complex case, there has been work done on polynomial approximation of analytic functions, defined on Runge or polynomially convex sets in infinite dimensional spaces. In the real case, there has been interest in the general problem of approximating, in one of several topologies, certain classes of differentiable functions by smoother ones, such as polynomials or real analytic functions. Some new results relating to approximation of differentiable functions with respect to four topologies are outlined in the chapter. The approximation of differentiable functions, in the fine topology, defined on C(K), where K is a compact Hausdorff space, is examined.
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