Linear Analysis (Michaelmas 2018)

Schedule

• 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.

Lecture Notes from 2018

Lecture Notes from 2017

Example Sheets

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

• Example Sheet 1 (posted October 5)
• Example Sheet 2 (posted October 19)
• Example Sheet 3 (posted November 2)
• Example Sheet 4 (posted November 16)

List of lectures

1. Normed vector spaces. The spaces l^p. Hoelder and Minkowski inequalities.
2. Banach spaces. Bounded operators.
3. Bounded operators. The dual space.
4. Finite-dimensional spaces. Completion.
5. Completion, products, quotients.
6. 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. Compact operators. Application of the Arzelà-Ascoli theorem: Peano's Existence Theorem.
12. Conclusion of Peano's Existence Theorem. Babylonian method to compute square roots.
13. Stone-Weierstrass Theorem.

References

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

• R. Bauerschmidt, Linear Analysis, Lecture Notes; available here.
• 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.