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1. Prove that c∗ is isometrically isomorphic to `1 . 2. Prove that if the unit ball in a normed space is compact then the space is finite dimensional. 3. Let {Xi }i∈I be a collection of normed spaces, and 1 ≤ p < ∞. Consider the set M p Xi i∈I consisting of sets {xi }i∈I such that xi ∈ Xi and X kxi kp < ∞. i Prove that this is a vector space and !1/p k{xi }i∈I k = X kxi kp i is a norm on it. 4. Let Y be a closed subspace of a normed space X. Show that there exist isometric isomorphisms Y∗ ∼ = X ∗ /Y ⊥ and (X/Y )∗ ∼ = Y ⊥. 5. Show that the space C[a, b] is not reflexive. 6. Let Y be a subspace of a normed space X. Show that Y ∗∗ can be identified with (Y ⊥ )⊥ ⊂ X ∗∗ . Conclude that if X is a reflexive Banach space then any closed subspace of X is also reflexive. 7. Let L be a Banach limit, that is, a linear functional on `∞ such that (i) kLk = 1; (ii) L(a) = limn an for a ∈ c; (iii) if an ≥ 0 for any n then L(a) ≥ 0; (iv) if a and a0 are such that a0n = an+1 then L(a) = L(a0 ). Compute L((1, 0, 1, 0, . . .)).