Functional Analysis Ii

1. Define and identify normed linear spaces and Banach spaces.
2. Define and identify Hilbert spaces and their properties.
3. Apply Holder's and Minkowski's inequalities.
4. Compute norms of bounded linear maps.
5. Understand and apply the Hahn-Banach theorem.
6. Understand and apply the Riesz representation theorem.
7. Define and identify adjoint, self-adjoint, normal, and unitary operators.

13 weeks, about 51 hours in total. Yours will differ.

1. Week 1Module 1: Normed Linear Spaces

   • Unit 1: Normed Linear Spaces · 3 hours

     Read the introduction and objectives of Unit 1.. Study the definitions of norm and normed linear space.. Work through Example 3.1 to verify the norm axioms on the real line R.. Practice applying the norm definitions to solve problems..

2. Week 2Module 1: Normed Linear Spaces

   • Unit 1: Normed Linear Spaces · 4 hours

     Study Example 3.2 and verify the norm conditions for R^2.. Understand Holder's and Minkowski's inequalities.. Review Definition 3.3 and examples of lp spaces.. Solve problems related to Holder's and Minkowski's inequalities..

3. Week 3Module 1: Normed Linear Spaces

   • Unit 1: Normed Linear Spaces · 4 hours

     Study examples 3.3, 3.4, and 3.5 to understand different norms on R^n and C^n.. Verify the norm axioms for the norms defined in these examples.. Practice applying these norms to solve problems in R^n and C^n..

4. Week 4Module 1: Normed Linear Spaces

   • Unit 1: Normed Linear Spaces · 4 hours

     Study examples 3.6, 3.7, 3.8, and 3.9 to understand norms on function spaces.. Verify the triangle inequality for the norms defined on C[a, b].. Understand the concept of equivalent norms and Theorem 3.1.. Solve problems related to equivalent norms and function spaces..

5. Week 5Module 1: Normed Linear Spaces

   • Unit 1: Normed Linear Spaces · 4 hours

     Study Proposition 3.6 and Example 3.11 to understand how norms induce metrics.. Define convex sets and convex functions.. Work through examples 4.1 to 4.6 to identify convex sets.. Solve problems related to convex sets and functions..

6. Week 6Module 2: Banach Spaces

   • Unit 2: Banach Spaces · 4 hours

     Read the introduction and objectives of Unit 2.. Review definitions of convergence and Cauchy sequences in metric spaces.. Understand the concept of a complete normed linear space.. Study Theorem 4.1 and its proof for the completeness of R^n and C^n..

7. Week 7Module 2: Banach Spaces

   • Unit 2: Banach Spaces · 4 hours

     Study Theorem 4.2 and its proof for the completeness of l∞.. Understand the completeness of C[a, b] with the sup norm.. Review examples of incomplete normed linear spaces.. Solve problems to identify complete and incomplete normed linear spaces..

8. Week 8Module 3: Linear Functional Spaces

   • Unit 3: Linear Functional Spaces · 4 hours

     Read the introduction and objectives of Unit 3.. Study the definition of linear maps and linear functionals.. Work through Example 3.1 to verify linearity of a map on l^2.. Understand Proposition 3.1 and its implications..

9. Week 9Module 3: Linear Functional Spaces

   • Unit 3: Linear Functional Spaces · 4 hours

     Understand the concept of bounded linear maps and Theorem 4.1.. Study Definition 4.3 and Theorem 4.2 to compute norms of bounded linear maps.. Review examples 4.1 and Proposition 4.1.. Solve problems related to bounded linear maps and their norms..

10. Week 10Module 3: Linear Functional Spaces

    • Unit 3: Linear Functional Spaces · 4 hours

      Understand the concept of dual or conjugate space.. Study Proposition 5.2 and Example 5.3 to understand dual spaces of lp^n.. Review the Hahn-Banach Theorem and its corollary.. Solve problems related to dual spaces and the Hahn-Banach Theorem..

11. Week 11Module 4: Inner Product Spaces

    • Unit 4: Inner Product Spaces · 4 hours

      Read the introduction and objectives of Unit 4.. Study the definition of inner product spaces and their basic properties.. Work through examples 3.1 to 3.5 to identify inner product spaces.. Understand Lemma 3.1 (Cauchy-Schwartz Inequality) and Theorem 3.1..

12. Week 12Module 4: Inner Product Spaces

    • Unit 4: Inner Product Spaces · 4 hours

      Understand Proposition 3.1 (Parallelogram Law) and Proposition 3.2 (Polarization Identity).. Review examples of inner product spaces.. Understand the Jordan Von Neumann Theorem.. Solve problems related to inner product spaces and the parallelogram law..

13. Week 13Module 5: Hilbert Spaces

    • Unit 5: Hilbert Spaces · 4 hours

      Read the introduction and objectives of Unit 5.. Study the definition of Hilbert spaces and their examples.. Understand orthogonality, orthonormal sets, and orthogonal complements.. Review the Projection Theorem and Direct Sum Decomposition..