## MAS6340 Analysis I

 Both semesters, 2021/22 20 Credits Lecturer: Dr Paul Mitchener Timetable Aims Outcomes 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.

There are no prerequisites for this module.
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

By the end of the unit, a candidate will be able to demonstrate the ability to, the capability for... 1. understand Banach spaces, Hilbert spaces, bounded linear maps and a range of examples; 2. understand orthonormal sequences in Hilbert spaces; 3. understand the Hahn-Banach theory; 4. understand the spectrum of an operator; 5. understand Fredholm operators and properties of the index;.
20 lectures, 20 tutorials

## Assessment

The module will be assessed by a formal, closed book, two and a half hour examination at the end of the second semester.

## 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 and Examples. 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. 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.
• Wavelets
Concept of a wavelet. The wavelet series. The integral wavelet transform. Time-frequency analysis. Inversion formulae and duals.