# Real and Functional Analysis: Part B: Functional Analysis by Arunava Mukherjea, K. Pothoven

By Arunava Mukherjea, K. Pothoven

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Let X be a complex Banach space, G an open set in the complex plane, and f a function from G into X. Then f is called analytic on G iff is differentiable at each point Ao E G, that is, II f(~ =1Ao) - J'(Ao) II ~ 0, as A ~ Ao for some function J': G ~ X. 3 above) to show that a necessary and sufficient condition for f to be differentiable on G is that x*(J(A» be differentiable on G for each x* E X*. Since the "necessary" part is trivial, we show only the "sufficient" part. Let x* (J(A) be differentiable on G and Ao E G.

Ji/ and T = {t/>A: A E fT}. Then the class S* of all continuous linear functionals on S is precisely the linear span of T. [This result is due to T. K. Mukherjee and W. H. Ji/ contains an atom. 18. Another Application of the Hahn-Banach Theorem. The classical moment problem can be stated as follows: Given a sequence of real numbers (an), when does there exist a real-valued function g of bounded variation on [0, 1] such that xn dg(x) = an, n = 0, 1, 2, ... Show that this problem can be answered in an abstract setup as follows: Let X be a normed linear space, (XA)A€A be elements in X, and (aA)A€A be scalars.

An Extension of the Hahn-Banach Theorem. Let p be a real-valued function of the linear space X over the reals such that p(x + y) < p(x) + p(y) and p(ax) = apex), if a >0. Supposefis a linear functional on a subspace S such that f(s) < pes) for all s E S. Suppose also that is an Abelian semigroup of linear operators on X (that is T I , T2 E ~ implies TIT2 = T2TI E ~) such that if T EfT, then p(T(x» < p(x) for all x E X andf(T(s» = f(s) for all s E S. Then there is an extension c/> off to a linear functional on X such that C/>(x) < p(x) and C/> (T(x) ) = C/>(x) for all x E X.