Linear Analysis (Michaelmas 2016)

Syllabus

• Normed and Banach spaces. Linear mappings, continuity, boundedness, and norms. Finite-dimensional normed spaces.
• The Baire category theorem. The principle of uniform boundedness, the closed graph theorem and the inversion theorem; other applications.
• The normality of compact Hausdorff spaces. Urysohn's lemma and Tiezte's extension theorem. Spaces of continuous functions. The Stone-Weierstrass theorem and applications. Equicontinuity: the Arzelà-Ascoli theorem.
• Inner product spaces and Hilbert spaces; examples and elementary properties. Orthonormal systems, and the orthogonalization process. Bessel's inequality, the Parseval equation, and the Riesz-Fischer theorem. Duality; the self duality of Hilbert space.
• Bounded linear operations, invariant subspaces, eigenvectors; the spectrum and resolvent set. Compact operators on Hilbert space; discreteness of spectrum. Spectral theorem for compact Hermitian operators.

Example Sheets

The example sheets will be posted here during the course of the term.

• Example Sheet 1 (posted October 4)
• Example Sheet 2 (posted October 18)
• Example Sheet 3 (posted November 1)
• Example Sheet 4 (posted November 15)

Miscellaneous

• Balanced bounded convex neighbourhoods

Classes

1. Normed and topological vector spaces. Definitions and basic properties. Topological vector spaces are normed iff locally convex and locally bounded.
2. Examples of normed and topological vector spaces. Equivalence of continuity and boundedness of linear maps on locally bounded spaces.
3. The Banach space of bounded linear maps. The dual and double dual. Examples.
4. Equivalence of norms on finite dimensional space; consequences. A normed space is finite dimensional iff its closed unit ball is compact. Statement of Hahn-Banach Theorem.
5. Proof of Hahn-Banach Theorem. Posets and Zorn's Lemma. Existence of bases.
6. Richness of the dual space. Baire Category Theorem.
7. Principle of uniform boundedness, Banach-Steinhaus Theorem. Open mapping theorem.
8. Conclusion of proof of open mapping theorem. Examples. Closed graph theorem. Normal topological spaces.
9. Urysohn-Tietze Extension Theorem.
10. Compactness in metric spaces. Arzelà-Ascoli theorem.
11. Application of the Arzelà-Ascoli theorem: Peano's Existence Theorem.
12. Babylonian method to compute square roots. Stone-Weierstrass Theorem.
13. Proof of Stone-Weierstrass Theorem.
14. Application of Stone-Weierstrass Theorem. Example to motivate weak topologies.
15. Banach-Alaoglu Theorem.
16. Application of Banach-Alaoglu Theorem: invariant measures for dynamical systems on compact metric spaces. Definition of Euclidean vector spaces and Hilbert spaces.
17. Examples of Euclidean vector spaces and Hilbert spaces. Orthogonality.
18. Riesz representation theorem. Orthogonal projections.
19. Orthonormal systems and isomorphism between separable Hilbert spaces and l². Definition of spectrum and resolvent.
20. Properties of spectrum and resolvent. Point spectrum, continuous spectrum, residual spectrum.
21. Spectrum of normal linear operators on a Hilbert space.
22. Spectral decomposition for compact self-adjoint operators.
23. Application of spectral theorem to boundary value problem.
24. Continuous functional calculus for bounded self-adjoint operators.

References

There are many classic textbooks on the subject. They can usually be found under the name Functional Analysis. However, I will mostly follow the following references used in previous years:

• B. Bollobás, Linear Analysis, An Introductory Course, Cambridge University Press; available here (from university network).
• M. Dafermos, Linear Analysis, Lecture Notes; available here.
• T.W. Körner, Linear Analysis, Lecture Notes; available here.