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).
|||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|
|||A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)|
|||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)|
Hahn–Banach theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hahn%E2%80%93Banach_theorem&oldid=22537