The Hahn-Banach Theorems
In a Hilbert space, we can introduce the notion of orthogonal coordinates through an orthogonal base, and these coordinates are the values of bounded linear functionals defined by the vectors of the base. This suggests that we consider continuous linear f
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IV. The Hahn-Banach Theorems
IV. The Hahn-Banach Theorems In a Hilbert space, we can introduce the notion of orthogonal coordinates through an orthogonal base, and these coordinates are the values of bounded linear functionals defined by the vectors of the base . This suggests that we consider continuous linear functionals, in a linear topological space, as generalized coordinates of the space. To ensure the existence of non-trivial continuous linear functionals in a general locally convex linear topological space, we must rely upon the Hahn-Banach extension theorems.
1. The Hahn-Banach Extension Theorem in Real Linear Spaces Theorem (HAHN [2], BANACH [1]). Let X be a real linear space and
let
p (x) be a real-valued function defined on X satisfying the conditions :
p(x + y)
~
p (x) + P(y)
P (IX x) = IXP (x) for
(1)
(subadditivity) , 1X:2:: O.
(2)
Let M be a real linear subspace of X and 10 a real-valued linear functional defined on M :
10(IX X + (Jy) = 1X10 (x) + {J 10 (y) for x, y EM and real (J . (3) Let 10 satisfy 10 (x) ~ P(x) on M. Then there exists a real-valued linear functional F defined on X such that i) F is an extension of 10 , i.e., (x,
F(x) = 10 (x) for all xE M, and ii) F(x) < P(x) on X. Proof. Suppose first that X is spanned by M and an element X o EM, that is, suppose that
X = {x = m
+ IXXO; mE M,
IX real}.
Since XoEM, the above representation of x E X in the form x = m is unique. It follows that, if, for any real number c, we set
+ IXXv
+
F(x) = F(m + IXXo) = lo(m) IXC , then F is a real linear functional on X which is an extension of 10 , We have to choose Csuch that F(x) ~p (x), thatis, 10 (m) IXC::;;; p(m + IXXo)' This condition is equivalent to the following two conditions:
+
+
lo(m/IX} C::;;; p (xo + m/IX) for IX> 0, 10 (m/(-tx)) - C::;;; P(-xo + m/(-tx)) for IX < O. To satisfy these conditions, we shall choose C such that 10 (m') -p(m' -xo) ~ C < p(m" + xo) - /0 (m") for all m', m" E M. Such a choice of c is possible since fo(m')
+ lo(m") =/o(m! + m")
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