2. [25pts] (a) [9 pts) Show that the dual space (Co)' is isometrically isomorphic to l' (here co is a subspace of 1 consisting of sequences of scalars converging to zero). P.T.O 2 ZEZ (b) [8 pts) State the Hahn-Banach theorem. Let Z be a subspace of a normed linear space X = (X, || - ||) and suppose 3 X is such that K = inf || X 2|| >0. Show there exists a bounded linear functional f on X with || | || k f(x) = 1 and f(-) = 0 Vz e Z. (c) Let U, V and W be nonempty subsets of an inner product space X with W CV. (i) [4 pts) Show V+ CW. (ii) [4 pts) Show = U. 2. [25pts] (a) [9 pts) Show that the dual space (Co)' is isometrically isomorphic to l' (here co is a subspace of 1 consisting of sequences of scalars converging to zero). P.T.O 2 ZEZ (b) [8 pts) State the Hahn-Banach theorem. Let Z be a subspace of a normed linear space X = (X, || - ||) and suppose 3 X is such that K = inf || X 2|| >0. Show there exists a bounded linear functional f on X with || | || k f(x) = 1 and f(-) = 0 Vz e Z. (c) Let U, V and W be nonempty subsets of an inner product space X with W CV. (i) [4 pts) Show V+ CW. (ii) [4 pts) Show = U