3103 (Functional Analysis)
Year: 2008–2009
Code: MATH3103
Old Code: MATHC336
Value: Half unit (= 7.5 ECTS credits)
Term: 2
Structure: 3 hour lectures per week.
Assessment: 100% examination
Normal Pre-requisites: 7102
Lecturer: o
Prof M Cs¨rnyei
Course Description and Objectives
The central concept of Function Analysis is the Banach space, which provides a unifying frame-
work for many important results in mathematics, and about which there is a well-developed
theory. Loosely speaking, a Banach space is a vector space together with a notion of a distance.
The distance allows one to define concepts like continuity and convergence in a more general
setting. In other words this course is, as its name suggests, analysis. The basic results of
Banach space theory will be presented, as well as some abstract analysis.
Detailed Syllabus
1. Metric spaces. Open and closed sets, boundary, interior. Continuus maps, homeomorphisms,
isometries. Connectedness, completeness, compactness. Properties equivalent to compact-
ness. Banach fixed point theorem for contraction mappings.
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2. Banach spaces. Examples: lp , lp , Lp , C[0, 1]. Proofs of completeness of these spaces.
3. All n-dimensional normed spaces are Banach spaces and equivalent to each other. All linear
maps between them continuous.
4. Zorn’s lemma, the Hahn-Banach theorem, applications.
5. Linear functionals and duality. Dual of lp is lq . Second dual and reflexive spaces.
6. Baire’s category theorem. Open mapping theorem, closed graph theorem, principle of uni-
form boundedness.
7. Hilbert spaces. Basic properties. Orthogonal systems and the orthogonalization process.
Representation of linear functions. Bounded linear operators on Hilbert spaces. Self-adjoint
and compact operators. Spectrum of an operator; classification of spectra.
May 2008 3103