Functional analysis II
MATH-404
Contents
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This page contains the tentative lecture contents.
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In
this course we treat selected topics in functional analysis.
§0. A short compendium of topology
§1. Locally convex topological vector spaces
- Vector space operations are continuous
- Existence of convex neighborhood basis
- Characterization with seminorms
- Metrizibilty and normability
- Dual spaces
§2. Test functions and distributions
- Smooth functions with compact support as locally convex TVS
- Distributions as the dual space
- Operations on distributions
- Schwartz space, tempered distributions, and the Fourier transform
§3. Calculus on Banach spaces
- Definition of derivatives and elementary properties
- Implicit function theorem
- Inverse function theorem
§4. Fixed point theorems
- Some theorems in the spirit of Banach's fixed point theorem
- Brouwer's fixed point theorem
- Schauder's fixed point theorem on locally convex TVS
§5. Introduction to degree theory (if time permits)
- Existence and uniqueness of the degree (analytical approach) in finite dimensions
- Some short proofs of deep theorems