Mathematics 421/510: current and upcoming topics

Chapter 1: Linear Spaces

Chapter 2: Linear Maps

• We only need the definition of a linear mapping, the basic properties at the beginning of Section 2.1, and Theorem 1 in that section. We are skipping the rest of the chapter.

Chapter 3: The Hahn-Banach Theorem

• The Hahn-Banach theorem
• The gauge of a convex body and the hyperplane separation theorem
• The Hahn-Banach theorem for complex linear spaces (Section 3.3, Theorem 8)
• Skip Theorem 7 in Section 3.3

Interlude: L^p Spaces

• The Holder and Minkowski inequalities
• L^p spaces as Banach spaces
• Approximation of L^p functions by simple functions (on general metric spaces) and continuous functions (on Rⁿ)
• C₀(Rⁿ) as the completion of C_c(Rⁿ)
• This material is not included in the Lax textbook. See any introductory text on real analysis, e.g. W. Rudin, "Real and Complex Analysis", or G. Folland, "Real Analysis".

Chapter 5: Normed Linear Spaces

• Normed spaces and Banach spaces
• Separability
• Noncompactness of the unit ball
• Uniform convexity

Chapter 6: Hilbert Spaces

• Scalar product, orthogonality, orthogonal decomposition
• Linear functionals on Hilbert spaces
• Orthonormal bases
• Application: Dirichlet's problem (Section 7.2)

Chapter 8: Duals of Normed Linear Spaces

• Bounded linear functionals and dual spaces
• L^p duality (see e.g. W. Rudin, "Real and Complex Analysis")
• Reflexive spaces

Chapter 10: Weak Convergence

• Weak convergence of sequences
• Principle of uniform boundedness (I followed the proof in Conway's Functional Analysis)
• Weak sequential compactness
• Weak* convergence of sequences
• Weak* sequential compactness

Chapter 11: Applications of Weak Convergence

• Approximate identity (Section 11.1)
• Divergence of Fourier series (Section 11.2)
• Weak solutions of partial differential equations (Section 11.5; we only covered the 1-dimensional case)

Chapter 12: The Weak and Weak* Topologies

• Weak topology, weak* topology
• Alaoglu's theorem

Chapter 15: Bounded Linear Maps

• Bounded linear maps, norm, nullspace, range, transpose
• Strong and weak convergence of operators
• The open mapping and closed graph theorems (Section 15.5; we'll skip the proofs)

Chapter 28: Compact Symmetric Operators in Hilbert Space

• For the definition of a compact operator, see Section 21.1.
• Symmetric operators on Hilbert spaces
• The spectral theorem for compact symmetric operators (Theorem 3)
• Skip the rest of the chapter.

Chapter 29: Examples of Compact Symmetric Operators

• The inverse of a differential operator (Section 29.2)

Conclusion: A very brief introduction to unbounded operators