Linear Topological Spaces

This book is a study of linear topological spaces. Explicitly, we Arc concerned with a linear space endowed with a topology such that scalar Multiplication and addition arc continuous, and wc scck invarlants relative To the class of all topological isomorphisms. Thus, from our point of view, It is incidental that the evaluation map of a normcd linear space nto its Second adjoint space is an isomctry; t is pertinent that this map is relatively Open. We study the geometry of a linear topological space for its own sake, And not as an ncidcntal to the study of mathematical objects which are Endowed with a more elaborate structure. This is not because the relation Of th{s theory to other notions is of no mportancc. On the contrary, any Discipline worthy of study must illuminate neighboring areas, and motivaTion for the study of a new concept may, n great part, lie in the clariication And slmplication of more familiar notions. As it turns out, the theory of Linear topological spaces provides a remarkable economy in discussion of Many classical mathemat{cal problems, so that this theory may properly be Considered to be both a synthesis and an cxtenslon of older ideas. The text begins with an invcstigat{on of linear spaces (not endowed with A topology). The structure hcrc is simple, and complete invariants :or a Space, a subspacc, a linear function, and so on, arc given in terms of cardinal Numbers. The geometry of convex sets s the irst topic which is peculiar to The theory of linear topological spaces. The fundamental propositions hcrc (the hahn-banach theorem, and the relation between ordcrings and convex Cones) yield one of the three general methods which are available for attack On linear topological space problems.