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
Classes
- Normed and topological vector spaces. Definitions and basic properties. Topological vector spaces are normed iff locally convex and locally bounded.
- Examples of normed and topological vector spaces. Equivalence of continuity and boundedness of linear maps on locally bounded spaces.
- The Banach space of bounded linear maps. The dual and double dual. Examples.
- 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.
- Proof of Hahn-Banach Theorem. Posets and Zorn's Lemma. Existence of bases.
- Richness of the dual space. 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.
- Application of the Arzelà-Ascoli theorem: Peano's Existence Theorem.
- Babylonian method to compute square roots. Stone-Weierstrass Theorem.
- Proof of Stone-Weierstrass Theorem.
- Application of Stone-Weierstrass Theorem. Example to motivate weak topologies.
- Banach-Alaoglu Theorem.
- Application of Banach-Alaoglu Theorem: invariant measures for dynamical systems on compact metric spaces. Definition of Euclidean vector spaces and Hilbert spaces.
- Examples of Euclidean vector spaces and Hilbert spaces. Orthogonality.
- Riesz representation theorem. Orthogonal projections.
- Orthonormal systems and isomorphism between separable Hilbert spaces and l^{2}. Definition of spectrum and resolvent.
- Properties of spectrum and resolvent. Point spectrum, continuous spectrum, residual spectrum.
- Spectrum of normal linear operators on a Hilbert space.
- Spectral decomposition for compact self-adjoint operators.
- Application of spectral theorem to boundary value problem.
- 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: