## 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.

## 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

- Normed vector spaces. The spaces l^p. Hoelder and Minkowski inequalities.
- Banach spaces. Bounded operators.
- Bounded operators. The dual space.
- Finite-dimensional spaces. Completion.
- Completion, products, quotients.
- Baire Category Theorem.
- Principle of uniform boundedness, Banach-Steinhaus Theorem. Open mapping theorem.
- Conclusion of proof of open mapping theorem. Examples. Closed graph theorem. Normal topological spaces.
- Urysohn-Tietze Extension Theorem.
- Compactness in metric spaces. Arzelà-Ascoli theorem.
- Compact operators. Application of the Arzelà-Ascoli theorem: Peano's Existence Theorem.
- Conclusion of Peano's Existence Theorem. Babylonian method to compute square roots.
- 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.