The Existence of Non-Trivial Bounded Functionals Implies the Hahn-Banach Extension Theorem | Zendy

W. A. J. Luxemburg | Zendy; Martin Väth | Zendy

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The Existence of Non-Trivial Bounded Functionals Implies the Hahn-Banach Extension Theorem

Author(s) -

W. A. J. Luxemburg,

Martin Väth

Publication year - 2001

Publication title -

zeitschrift für analysis und ihre anwendungen

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.567

H-Index - 35

eISSN - 1661-4534

pISSN - 0232-2064

DOI - 10.4171/zaa/1015

Subject(s) - extension (predicate logic) , mathematics , bounded function , pure mathematics , banach space , bounded inverse theorem , eberlein–šmulian theorem , discrete mathematics , mathematical analysis , finite rank operator , lp space , computer science , programming language

We show that it is impossible to prove the existence of a linear (bounded or unbounded) functional on any L_∞/C_0 without an uncountable form of the axiom of choice. Moreover, we show that if on each Banach space there exists at least one non-trivial bounded linear functional, then the Hahn-Banach extension theorem must hold. We also discuss relations of non-measurable sets and the Hahn-Banach extension theorem.

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