Mathematics 421/510: current and upcoming topics

Linear Spaces and Linear Maps

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

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)
• All the necessary background is contained in either W. Rudin, "Real and Complex Analysis", or G. Folland, "Real Analysis". If you are using Rudin, Chapter 1 and 2 give the general background on integration. We will only work with "regular" measures (Rudin, Theorem 2.18). Chapter 3 covers all the necessary background on L^p spaces.

Normed Linear Spaces

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

Hilbert Spaces

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

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

Weak Convergence

• Weak convergence of sequences
• Principle of uniform boundedness
• Weak sequential compactness
• Weak* convergence of sequences
• Weak* sequential compactness

Applications of Weak Convergence

• Approximate identity (Lax, Section 11.1)
• Divergence of Fourier series (Lax, Section 11.2)
• Weak solutions of partial differential equations (Lax, Section 11.5)

The Weak and Weak* Topologies

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

Bounded Linear Maps

• Bounded linear maps, norm, nullspace, range, transpose
• Strong and weak convergence of operators
• The open mapping and closed graph theorems

Compact Symmetric Operators in Hilbert Space

• Symmetric operators on Hilbert spaces
• The spectral theorem for compact symmetric operators

Examples of Compact Symmetric Operators

• The inverse of a differential operator (Lax, Section 29.2)

Conclusion: A very brief introduction to unbounded operators