Master Synthesis Exam – Functional Analysis

Comprehensive Exam: Functional Analysis 🧮

Level: Master 2 Mathematics

This comprehensive exam brings together the essential concepts of Functional Analysis at the Master 2 level and evaluates your mastery of advanced concepts. It is designed to test both your theoretical understanding and your ability to solve complex analytical problems.

🔎 Objectives

The exam covers the following topics:

• Continuous Linear Functionals: Demonstrate that a function f is linear and continuous, and calculate its norm.
• Non-Equivalent Norms: Identify and prove that two norms on a given vector space are not equivalent.
• Completeness of the Space: Verify whether this space is complete and understand its topological properties.
• Applications of the Ascoli-Arzelà Theorem: Study the uniform convergence and compactness of sequences of continuous functions.

📘 Typical Content

• Theoretical exercises to consolidate advanced definitions and properties.
• Practical problems involving calculations, proofs, and applications of theorems.
• Critical analysis of sequences and transformations in normed and functional spaces.