Hahn-Banach theorem
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).
References
[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) |
Comments
A real-valued function is called subadditive if
for all
in its domain such that
lies in its domain.
References
[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://encyclopediaofmath.org/index.php?title=Hahn-Banach_theorem&oldid=22537