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7.5.1 Compact Self-Adjoint Operators; The Spectral Theorem . Functional analysis is an abstract branch of mathematics that originated from classicalFunctional analysis takes up topological linear spaces, topological groups, normed rings, modules of representations of topological groups in topological linear 5.2 Hahn-Banach extension theorem – a second version . Functional analysis is the analysis of infinite dimensional vector spaces, typically spaces of theorem has a number of interesting applications to problems in analysis, but since our concern here is mainly with matters of algebra and topology, Three Basic Facts in Functional Analysis. 17. 8. The Hahn-Banach Extension Theorem. 20. 9. Dual Spaces. 23. 10. Weak Convergence and Eberlein's Theorem.