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Semilinear (Topological) Spaces and Applications
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Excerpt from Semilinear (Topological) Spaces and Applications: June, 1971 In this paper we define semivector spaces and develop some of their algebraic aspects including some structure theory; then topopogize these spaces to obtain semilinear topological spaces for which we identify a hierarchy of local convexity axioms; establish a number of fixed point and minmax theorems for spaces with various local convexity properties; illustrate how the spaces of concern arise naturally as various hyperspaces of linear and semilinear (topological) spaces; and, finally, indicate briefly how all the above are applied in socio-economic analysis and optimization. In contrast to vector spaces, we build semivector spaces upon underlying commutative semi-groups, so that the space need not have an origin (identity element) and its elements need not possess inverses. Neither is Os required to equal the origin e even if e belongs to the semivector space (0 denotes the additive identity of the field). Also, in the case where the reals are taken as the field, singleton sets need not be convex. These are some of the features distinguishing semivector spaces from vector spaces. Section 1 treats these and other algebraic matters pertaining to semivector spaces. As a result, some ("gross") structure theory is also developed. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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