This chapter is devoted to the algebra and the geometry of linear spaces; no topology for the space is assumed. It is shown that a linear space is determined, to a linear isomorphism, by a single cardinal number, and that subspaces and linear functions can be described in equally simple terms. The structure theorems for linear spaces are valid for spaces over an arbitrary field; however, we are concerned only with real and complex linear spaces, and this restriction makes the notion of convexity meaningful. This notion is fundamental to the theory, and almost all of our results depend upon propositions about convex sets. In this chapter, after establishing connections between the geometry of convex sets and certain analytic objects, the basic separation theorems are proved. These theorems provide the foundation for linear analysis; their importance cannot be overemphasized.
KeywordsLinear Space Linear Subspace Null Space Linear Isomorphism Real Linear Space
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