Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
|Each topic requires approximately 2 weeks to cover||Chapters 1, 2, and 4-6.||
|Metric and normed spaces (review): Metrics, norms, limits, liminf, limsup; Pointwise, uniform, and norm convergence; Continuity and completeness; Compactness in finite-dimensions; Compact and locally compact spaces.|
|Spaces of continuous functions: Definition of spaces; Convergence in the uniform topology; Tychonoff's Theorem; Arzela-Ascoli Theorem; Stone-Weierstrass Theorem|
|Topological spaces: Definition of topological spaces; Bases of open sets; Comparing topologies|
|Banach spaces: Normed vector spaces; Linear functionals and bounded linear maps; The kernel and range of linear maps; Convergence in the space of bounded linear operators; Dual spaces|
|Hilbert spaces: Inner products; Orthogonality and projections; Orthonormal bases; Applications|