Question
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)*)
Solution Preview
This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.