Geometric Aspects of General Topology pp 71-131 | Cite as

# Topology of Linear Spaces and Convex Sets

Chapter

## Abstract

In this chapter, several basic results on topological linear spaces and convex sets are presented. We will characterize finite-dimensionality, metrizability, and normability of topological linear spaces. Among the important results are the Hahn–Banach Extension Theorem, the Separation Theorem, the Closed Graph Theorem, and the Open Mapping Theorem. We will also prove the Michael Selection Theorem, which will be applied in the proof of the Bartle–Graves Theorem.

## Keywords

Linear Space Scalar Multiplication Normed Linear Space Affine Function Neighborhood Basis
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## References

- 1.M. Eidelheit, S. Mazur, Eine Bemerkung über die Räume vom Typus (
*F*). Stud. Math.**7**, 159–161 (1938)MATHGoogle Scholar - 2.E. Michael, Continuous selections, I. Ann. Math.
**63**, 361–382 (1956); Continuous selections, II. Ann. Math.**64**, 562–580 (1956); Continuous selections, III. Ann. Math.**65**, 375–390 (1957)Google Scholar - 3.E. Michael, A generalization of a theorem on continuous selections. Proc. Am. Math. Soc.
**105**, 236–243 (1989)MathSciNetCrossRefMATHGoogle Scholar

## Copyright information

© Springer Japan 2013