Problem: Let X be a Banach space, and let X* denote its dual space, i.e., the space of all continuous linear functionals on X. Consider the following questions: 1. Hahn-Banach Theorem: Let YC X be a subspace and fЄ Y, where Y is the dual space of Y. Using the Hahn-Banach extension theorem, show that there exists F € X* such that F❘y = f and ||F|| = ||ƒ ||. 2. Weak Topology:* Define the weak* topology on X* as the coarsest topology such that for each xЄ X, the map or: X* → C given by oz (F) = F(x) is continuous. Prove that the closed unit ball of X, denoted Bx. = {F € X : ||F|| ≤ 1}, is compact in the weak* topology when X is a reflexive Banach space. 3. Banach-Alaoglu Theorem: Prove the Banach-Alaoglu theorem, which states that the closed unit ball Bx. of the dual space X* is compact in the weak* topology, even when X is not necessarily reflexive. 4. Reflexivity and Weak Convergence: Suppose X is a reflexive Banach space. Let {x} be a bounded sequence in X. Prove that there exists a subsequence {n,} which converges weakly to some a € X, i.e., lim f(x) = f(x) for all ƒ € X*. k→∞ Why is reflexivity essential in this result? 5. Duality Mapping and Uniform Convexity: Let X be a uniformly convex Banach space. Define the duality mapping J : X → 2** by J(x) = {F € X* : ||F|| = ||x|| and F(x) = ||x||²}. Prove that if X is uniformly convex, the duality mapping J is single-valued, norm-to-weak* continuous, and that X is reflexive.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

Hahn-Banach Extension: Use the geometric form of the Hahn-Banach theorem to extend the functional fff from the subspace Y to the whole space X without increasing its norm.
Weak Compactness:* For the second part, apply the Banach-Alaoglu theorem (which is discussed in part 3) to show that in reflexive spaces, the weak* topology on the dual space has compact unit balls.
Banach-Alaoglu Theorem: For the Banach-Alaoglu theorem, use Tychonoff's theorem on the product topology of infinite-dimensional spaces to prove that the dual unit ball is weak* compact.
Weak Convergence: Use the Eberlein-Šmulian theorem, which asserts that every bounded sequence in a reflexive Banach space has a weakly convergent subsequence.
Uniform Convexity: For the duality mapping, use the fact that uniform convexity implies the uniqueness of elements in the dual and reflexivity of the space, which gives rise to continuity properties of the mapping.