## MAS436 Functional Analysis

Note: This is an old module occurrence.

You may wish to visit the module list for information on current teaching.

 Both semesters, 2017/18 20 Credits Lecturer: Dr Paul Mitchener Home page Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

Functional analysis is the study of infinite-dimensional vector spaces equipped with extra structure. Such spaces arise naturally as spaces of functions. As well as being a beautiful subject in its own right, functional analysis has numerous applications in other areas of both pure and applied mathematics, including Fourier analysis, study of the solutions of certain differential equations, stochastic processes, and in quantum physics. In this unit we focus mainly on the study of Hilbert spaces- complete vector spaces equipped with an inner product- and linear maps between Hilbert spaces. Applications of the theory considered include Fourier series, differential equations, index theory, and the basics of wavelet analysis.

Prerequisites: MAS220 (Algebra); MAS331 (Metric Spaces); MAS332 (Complex Analysis)
No other modules have this module as a prerequisite.

## Outline syllabus

• Normed and Banach Spaces
• Linear Maps and Continuity
• Spaces of Continuous Functions
• Hilbert Spaces
• Orthonormal Sets
• Spectral Theory
• Fredholm Operators
• Wavelets

## Aims

• To introduce students to the ideas and some of the fundamental theorems of functional analysis.
• To show students the use of abstract algebraic/topological structures in studying spaces of functions.
• To allow students to taste the subject with a view to further work as a postgraduate.
• To give students a working knowledge of the basic properties of Banach spaces, Hilbert spaces and bounded linear operators.
• To show students the idea of duals and adjoints.
• To show students the value of looking at the spectrum of a bounded linear operator.
• To demonstrate significant applications of the theory of functional analysis.

## Learning outcomes

On successful completion of this course, students will be able to:
• Appreciate how functional analysis uses and unifies ideas from vector spaces, the theory of metrics, and complex analysis.
• Understand and apply fundamental theorems from the theory of normed and Banach spaces, including the Hahn-Banach theorem, the open mapping theorem, the closed graph theorem, and the Stone-Weierstrass theorem.
• Appreciate the role of Zorn's lemma.
• Understand and apply ideas from the theory of Hilbert spaces to other areas, including Fourier series, the theory of Fredholm operators, and wavelet analysis.
• Understand the fundamentals of spectral theory, and appreciate some of its power.

## Teaching methods

Independent reading, discussion classes, problem solving

40 lectures, no tutorials

## Assessment

One formal 2.5 hour written examination.

## Full syllabus

• Normed Spaces
Norms. Banach spaces and Completeness. Examples, including the spaces Lp [0,1].
• Linear Maps and Continuity
Bounded linear maps. Normed spaces of linear maps. The open mapping and closed graph theorems.
• Spaces of Continuous Functions
Dual Spaces. Zorn's Lemma. The Hahn-Banach theorem. The space of continuous functions on a compact metric space and the Stone-Weierstrass theorem.
• Hilbert Spaces
Inner product spaces. Associated norms, and the Cauchy-Schwarz inequality. Orthogonal complements and direct sums. Representation of functionals on Hilbert spaces, and adjoints of operators.
• Orthonormal Sets
Orthonormal sets and sequences, and related results. Application to Fourier series and Legendre polynomials.
• Spectral Theory
The spectrum of an operator. Complex analysis on Banach spaces. Non-emptiness and compactness of the spectrum. Self-adjoint and unitary operators. The spectral radius formula, and the spectral mapping theorem for polynomials.
• Compact Operators
Definition and basic properties of compact operators. The spectral theorem for compact self-adjoint operators. Application to differential equations.
• The Fredholm Index
Definition of a Fredholm operator and its index. Atkinson's theorem. Invariance properties of the index. Hardy spaces and Toeplitz operators. The Toeplitz index theorem.
• Fourier and Wavelets Analysis
The Fourier transform as and its properties. View of the Fourier transform as a unitary operator. Concept of a wavelet. The wavelet series. The integral wavelet transform. Inversion formula.