Functional Analysis Problems

1. Let X be a norm space and A be a subset of X. Let A⊥ = {f ∈ X* : f|A = 0 }.
Prove that A⊥ is a norm closed linear subspace of X*.

2. Let X be a norm space and B be a subset of X*. Let B⊥ = {x ∈ X : f(x) = 0 for all f ∈ B}. Prove that B⊥ is a norm closed linear subspace of X.

3. Recall that (c₀)* ≅ l₁ and (l₁)* ≅ l∞. Let M = {(xₙ) ∈ l¹ : x₁ = x₂).
(i) For each x = (xₙ) ∈ l¹, describe the coset x + M (as an element in l₁/M).
(ii) Find the subspace M⊥ of c₀.
(iii) Find the subspace M⊥ of l∞.

4. Let X be a normed space and A ⊆ X. Prove that
(i) A ⊆ (A⊥)⊥;
(ii) A = (A⊥)⊥ if A is a norm closed linear subspace of X. (Hint: use H-B theorem)

5*. Let M be a norm closed linear subspace of a reflexive Banach space X. Prove that M is also reflexive.
(Hint: show that = M = (M⊥)⊥ ≅ (M⊥)⊥ ≅ (X* / M⊥)* ≅ (M*)*,
and the identification M ≅ M** derived above is exactly the canonical embedding M --> M**, x --> x̄

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