Difference between revisions of "Math 540: Linear Analysis"
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Revision as of 16:39, 3 April 2013
Contents
Catalog Information
Title
Linear Analysis.
(Credit Hours:Lecture Hours:Lab Hours)
(3:3:0)
Recommended
Math 342 or equivalent.
Description
Normed vector spaces and linear maps between them.
Desired Learning Outcomes
The course is designed to cover elementary abstract linear functional analysis. "Elementary" means that methods dependent on complex analysis or measuretheoretic integration are not core topics. "Abstract" means that applications to specific function spaces are not core topics.
Prerequisites
The official prerequisite is Math 342. What's important is that incoming students be familiar with linear algebra and metric spaces and be mathematically mature.
Minimal learning outcomes
Students should obtain a thorough understanding of the topics listed below. In particular they should be able to define and use relevant terminology, compare and contrast closelyrelated concepts, and state (and, where feasible, prove) major theorems.
 Normed spaces
 Basics
 Banach spaces
 Special linear operators
 Continuous/bounded
 Compact
 Finite rank
 Duality
 Dual spaces
 Their completeness
 Adjoints of bounded linear operators
 Second duals
 Reflexivity
 Weak and weakstar topologies
 BanachAlaoglu theorem
 Dual spaces
 Structure
 Hamel and Schauder bases
 Biorthogonal systems
 Separability
 Direct sums
 Quotient spaces
 Finitedimensional spaces
 Equivalence of all norms
 Completeness
 Continuity of all linear operators
 Characterization: unit ball is compact
 Fundamental theorems
 Baire category theorem
 HahnBanach extension theorem
 BanachSteinhaus theorem
 Open mapping theorem
 Closed graph theorem
 Bounded inverse theorem
 Basics
 Inner product spaces
 Basics
 Hilbert spaces
 Special linear operators
 Selfadjoint
 Unitary
 Normal
 Orthogonal projections
 HilbertSchmidt operators
 Structure
 Orthogonality
 Complements and direct sums
 Bases
 Representation theorems
 RieszFrechet theorem
 LaxMilgram theorem
 Abstract Fourier theory
 RieszFischer theorem
 Bessel’s inequality
 Parseval’s identities
 Orthogonality
 Basics
 Spectral theory
 Banach algebras
 Bounded operators on Banach spaces
 Gelfand’s spectralradius formula
 Compact operators on Banach spaces
 RieszSchauder theory including Fredholm Alternative
 Compact normal operators on Hilbert spaces
 Compact selfadjoint operators on Hilbert spaces
Textbooks
Possible textbooks for this course include (but are not limited to):
 David Promislow, A First Course in Functional Analysis, Wiley, 2008.
Additional topics
While the focus of the course is on abstract theory, this theory should probably be motivated and illustrated with appropriate concrete examples.
Courses for which this course is prerequisite
This course is recommended for Math 640. Indirectly (through the Math 640), this course will possibly become be a prerequisite for Math 647.