Semi-norms

Abstract

The semi-norm of a vector in a linear space gives a kind of length for the vector. To introduce a topology in a linear space of infinite dimension suitable for applications to classical and modern analysis, it is sometimes necessary to make use of a system of an infinite number of semi-norms. It is one of the merits of the Bourbaki group that they stressed the importance, in functional analysis, of locally convex spaces which are defined through a system of semi-norms satisfying the axiom of separation. If the system reduces to a single semi-norm, the corresponding linear space is called a normed linear space. If, furthermore, the space is complete with respect to the topology defined by this semi-norm, it is called a Banach space. The notion of complete normed linear spaces was introduced around 1922 by S. Banach and N. Wiener independently of each other. A modification of the norm, the quasi-norm in the present book, was introduced by M. Fréchet. A particular kind of limit, the inductive limit, of locally convex spaces is suitable for discussing the generalized functions or the distributions introduced by L. Schwartz, as a systematic development of S. L. Sobolev’s generalization of the notion of functions.