Frölicher A., Kriegl A. Linear Spaces and Differentiation Theory

New York: John Wiley & Sons, 1988. — 131 p.This book presents a new basis for differential calculus. Classical differentiation in linear spaces of arbitrary dimension uses Banach spaces—but most function spaces are not Banach spaces. Any attempts to develop a theory of differentiation covering non-normable linear spaces have always involved arbitrary conditions. This book bases the theory of differentiability of linear spaces on the fundamental idea of reducing the differentiability of general maps to that of functions on the real numbers. And the property “continuously differentiable” is replaced by that of “Lipschitz differentiable.” The result is a more natural theory, of conceptual simplicity that leads to the the same categories of linear spaces, but in a more general setting.