# Hahn-Banach theorem

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Let be a linear manifold in a real or complex vector space . Suppose is a semi-norm on and suppose is a linear functional defined on which satisfies (*)

for every . Then can be extended to a linear functional on all of such that for all . Such is an extension is not uniquely determined.

In the case of a real space the semi-norm can be replaced by a positively-homogeneous subadditive function, and the inequality (*) by the one-sided inequality , which remains valid for the extended functional. If is a Banach space, then for one can take , and then . The theorem was proved by H. Hahn (1927), and independently by S. Banach (1929).

How to Cite This Entry:
Hahn-Banach theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hahn-Banach_theorem&oldid=15516
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article