# Linear Analysis (Michaelmas 2018)

Time and Location: M-W-F, 9-10am; room MR3

Linear Analysis is an introductory Functional Analysis course, in Part II of the Cambridge Tripos.

## 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.

## 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.