Normed Linear Spaces, 3rd Edition (Ergebnisse der Mathematik by Mahlon Marsh Day

By Mahlon Marsh Day

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For existence of a continuous linear extension of a function on a subset we have an analogue of Lemma I, 2,1. Theorem 2. ~ tJ(x i) . ~ tiXil/ l=n xn of X. 44 Chapter II. Normed Linear Spaces Proof. By Lemma I, 2, 1, f has a linear extension g defined on the linear set L of all linear combinations of points of X. By the hypothesis Ig(y)I~Mllyll if YEL; by the Hahn-Banach theorem, with Mllxll for p(x), g has an extension F with IIFII ~M. For a finite set X this yields Corollary 1. If Xl, ... fll ~M if and only if I i~}iCil ~MII i~}iXi I for all chOices of t 1, •..

Then tx+(1-t)YEC if and only if t>O. For discussions of the operation of lineal closure see Klee [5] and Nikodym. (11) If W is a wedge in Land W" is the polar set in L*, then (a) W"= {f:fEL* and f(x)~O for all x in W}. (b) W" is a w*-closed wedge in L*. (12) Tukey showed that two closed convex disjoint sets in a reflexive Banach space can be separated by a closed hyperplane. Dieudonne [5] showed that this property fails in P(OJ). Klee [2] shows how local compactness is needed in the separation theorem when no interior point is available.

4,2). (6) Theorem 6 is a simple consequence of the separation and support theorems. (7) If W has an interior point and f is a non-zero element of L # which is non-negative on W, then f(x»O at every interior point x of Wand fEL*. (8) The Hahn-Banach theorem follows directly from Theorem 6. [Let M=L xR, E={{x,fo(x)):XELo}. K={(x,r):r~p(x)}, W=K-E. Then the core of K is {(x, r): r > p(x)} so (0, 1) is a core point of K and of W. (0,0) is not a core point of W. Theorem 6 gives a non-trivial monotone F in M #.

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