Functional Analysis Problems

Question

1. Prove that if X and Y are normal spaces and A: X --> Y is a linear transformation, then A is closed if and only if whenever Xₙ --> 0 and AXₙ --> y, it must be that y = 0.

2. Suppose that II . II, and II . II₂ are two norms on a linear space X such that (x, || . ||) and (x, || . ||₂) are both Banach spaces. Prove that if ||x||₁ ≤ ||x||₂ for all x ∈ X, then || . ||, and || . ||₂ are equivalent norm on X.

3. Let X be a Banach space and the norm closed linear subspaces M and N are complemented in X. Then each k in X can be uniquely expressed by k = m + n with M ∈ M and n ∈ N. Prove that X/N --> M, X + N --> m is a linear homeomorphism.

4. If {kₙ} ⊆ l', then Σ⁰ⱼ₌₁ xₙ(j) y(j) --> 0 for every y in C₀ if and only if sup||xₙ|| < ∞ and xₙ(j) --> 0 for every j ≥ 1 (use the fact that l' ≅ (C₀)*)

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