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).


[1] H. Hahn, "Ueber lineare Gleichungsysteme in linearen Räume" J. Reine Angew. Math. , 157 (1927) pp. 214–229
[2a] S. Banach, "Sur les fonctionelles linéaires" Studia Math. , 1 (1929) pp. 211–216
[2b] S. Banach, "Sur les fonctionelles linéaires II" Studia Math. , 1 (1929) pp. 223–239
[3] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[4] L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian)


A real-valued function is called subadditive if for all in its domain such that lies in its domain.


[a1] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[a2] G. Köthe, "Topological vector spaces" , 1 , Springer (1969)
How to Cite This Entry:
Hahn-Banach theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article