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. Problem: Analysis of Linear Operators on Banach Spaces Context: Let X be a Banach space over K (where K is R or C), and let T : XX be a bounded linear operator. Suppose I satisfies the following conditions: 1. T is compact. 2. T is self-adjoint (if X is a Hilbert space; if X is a general Banach space, assume I satisfies (Tx, y) = (x, Ty) for all x, y in X, where (•, •) denotes the dual pairing). 3. The spectrum σ (T) of T consists only of 0 and a sequence {\n}₁ of non-zero real numbers converging to 0. Questions: Part A: Spectral Properties 1. Eigenvalues and Eigenvectors: Show that each X is an eigenvalue of T. • Prove that the corresponding eigenvectors {n} can be chosen to form an orthonormal (if X is a Hilbert space) or a Schauder (if X is a Banach space) basis for X. 2. Multiplicity: Demonstrate that each non-zero eigenvalue X, has finite multiplicity.