Advanced Functional Analysis Mastery Quiz Instructions: No partial credit will be awarded: any mistake will result in a score of 0. Submit your solution before the deadline. Ensure your solution is detailed, and all stops are well-documented. No Al tools (such as ChatGPT or others) may be used to assist in solving the problems. All work must be your own. Solutions will be checked for Al usage and plagiarism. Any detected violation will result in a score of 0. Problem Let X and Y be Banach spaces, and let T: X →Y be a bounded linear operator. Consider the following tasks: 1. [Banach Fixed-Point Theorem] a State and prove the Banach Fixed-Point Theorem (Contraction Mapping Theorem). Provide a detailed explanation of how the theorem guarantees the existence of a unique fixed point for a contraction mapping on a complete metric space. b. Let T: X → X be a contraction mapping on X = R² with T(r. u) = (3.). Find the unique fixed point of T. 2. [Duality and the Hahn-Banach Theorem] a. State the Hahn-Banach Theorem for extending a continuous linear functional from a subspace to the entire Banach space. Prove the theorem. b. Using the Hahn-Banach Theorem, show that for any bounded linear functional p on a Banach space X, there exists a unique az X such that (20) = ||||, and ||0|| = 1. 3. [Compact and Bounded Linear Operators] a. Define a compact operator on a Banach space X. Prove that every finite-rank operator is a compact operator. b. Show that the image of a bounded set under a compact operator is relatively compact, i.e. the closure of the image of a bounded set is compact. 4. [Fredholm Operators and Index Theorem] a. Define a Fredholm operator T : X → Y between Banach spaces. Prove that a Fredholm operator has finite-dimensional kernel and cokemel. b. Prove the Fredholm Index Theorem: If T is a Fredholm operator, then the index of T defined as index(T) = dim(ker(T)) dim(coker(T)), is constant on the set of invertible perturbations of 'T. 5. [Spectral Theory for Compact Operators] a. State and prove the Spectral Theorem for compact self-adjoint operators on a Hilbert space. Explain how this theorem implies that the spectrum of a compact self-adjoint operator consists only of eigenvalues and accumulates only at 0. b. Let T': X -> X be a compact linear operator on a Banach space X with ||T|| < 1. Prove that I I' is invertible and find a bound for (IT)¹||.