Convexity in Linear Topological Spaces
We have made good progress in developing the algebraic aspects of our subject but the needs and applications of functional analysis require more powerful methods based on topological concepts. Thus, as our next step, we consider the result of imposing on a given linear space a “compatible topology”. This is hardly a novel idea; indeed, several excellent books already exist which are devoted to a detailed investigation of the many ramifications of this notion. However, our treatment is less ambitious and more pragmatic, being shaped primarily by the necessities of our intended applications. These necessities require an understanding of the properties of topologies defined by one or more semi-norms on a linear space. They also require a well-rounded duality theory and it is interesting to discover that the maximal class of linear topologies which yields the requisite duality theory is precisely the class of topologies defined by a family of semi-norms.
KeywordsLinear Space Extreme Point Convex Subset Weak Topology Convex Space
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