Prove that l^(infinity), the set of all bounded sequences equipped with the sup-norm, is not separable. Hint: think of the decimal expansion proof that the real numbers are not countable. ********************************************************* A slight reformulation of the Hahn-Banach Theorem states that if L is a linear subspace of a normed space X and f:L -> R is a linear continuous function, then f has a linear continuous extension F:X -> R with ||F||=||f||, where ||f|| denotes the norm of f in the dual of L. (i) Explain why this follows from the Hahn-Banach Theorem (ii) Prove that this F is unique if H is a Hilbert space. ********************************************************* Refering to Theorem 1 on http://www.few.vu.nl/~jhulshof/FA2008/week5/DSC01950.JPG, let X be l^(infinity), the set of all bounded sequences equipped with the sup-norm, and give an example of an x for which the linear continuous map F:X -> R is not unique. ********************************************************* Refering to the proof of the Lemma on http://www.few.vu.nl/~jhulshof/FA2008/week5/DSC01951.JPG, show that the norm of f in the dual of the space spanned by L and x_0 is equal to 1. ********************************************************* Refering to the proof of Thm 2 on http://www.few.vu.nl/~jhulshof/FA2008/week5/DSC01951.JPG, (i) Prove that if X is a separable normes space, then the unit sphere {x in X: ||x||=1 } contains a countable set with closure equal to the unit sphere. (NB. In the proof (i) is used for the dual of X.) (ii) Assuming the claim in step 2 to be false, the lemma implies the existence of an F which is zero in every x_n and has ||F||=1. Derive a contradiction. ********************************************************* Optionary: refering to the proof of Hahn-Banach's Theorem for separable normed spaces on http://www.few.vu.nl/~jhulshof/FA2008/week5/DSC01947.JPG, show that alpha_1 can indeed be chosen as claimed. ********************************************************* Exercises related to steps in the proof of the closed graph theorem: Let K be a convex subset of normed space which is symmetric (-x in K if x is in K). Suppose that K has nonempty interior. Prove that K contains an open ball centered in the origin. Let X be a Banach space and x_n in X. If the sum over n of ||x_n|| is finite show that the sum over n of x_n exists in X (i.e. x_1+...+x_n converges to a limit as n -> infinity). Formulate and prove the obvious infinite triangle inequality for the sum of the series. *********************************************************** Banach-Steinhaus Theorem. Let X and Y be Banach spaces and let S be a subset of B(X,Y), the Banach space of all linear bounded operators from X to Y. Suppose that, for every x in X, the set Sx={Ax: A in S} is bounded. Prove that S is bounded in B(X,Y), i.e. {||A||, A in S} is bounded. Hint: let W_n={x in X: ||Ax|| < n for all A in S} Is it necessary for Y to be oomplete? Apply this theorem and the relation between X and its second dual to show that weakly convergent sequences are bounded. Corrollary: prove that weakly convergent sequences are bounded. ***********************************************************