On the connection between weak and strong convergencies in Cm

AbstractIn this work wome connections are pursued between weak and strong convergence in the spaces Cm (m-times continuously differentiable functions on Rn). Let fn, f ϵ Cm + 1, where n = 1, 2,…, and m is a nonnegative integer. Suppose that the sequence {fn} converges to f relative to the weak topology of Cm + 1. It is shown that this implies the convergence of {fn} to f with respect to the strong topology of Cm. Several corollaries to this theorem are established; among them is a sufficient condition for uniform convergence. A stronger result is shown to exist when the sequence constitutes an output sequence of a linear weakly continuous operator