# 5E: Functional Analysis MATHS5069

**Academic Session:**2018-19**School:**School of Mathematics and Statistics**Credits:**20**Level:**Level 5 (SCQF level 11)**Typically Offered:**Semester 2**Available to Visiting Students:**No**Available to Erasmus Students:**No

#### Short Description

This is a first course in functional analysis, i.e. the theory of infinite-dimensional vector spaces and of the linear maps between them. In order to be able to discuss the most relevant examples of infinite-dimensional function spaces, the course will begin with a concise development of the Lebesgue integral on R^n and more general spaces.

It will go on to study basic examples of Banach spaces related to the Lebesgue integral, othonormal bases in Hilbert and inner product spaces, basic operator and spectral theory.

#### Timetable

34 x 1 hr lectures and 12 x 1 hr tutorials in a semester

#### Requirements of Entry

Mandatory Entry Requirements

3H Analysis of Differentiation and Integration (MATHS4073)

3H Metric Spaces and Basic Topology.(MATHS4077)

Recommended Entry Requirements

3H Methods of Complex Analysis (MATHS4075)

#### Excluded Courses

4H: Functional Analysis (MATHS4103)

#### Assessment

Assessment

90% Examination, 10% Coursework.

Reassessment

Resit opportunities for MSc students.

**Main Assessment In:** April/May

#### Course Aims

This course aims to introduce students to the theory of Banach and Hilbert spaces, that is, infinite-dimensional vector spaces equipped with a norm respectively inner product that turns them into a complete

metric space, and of the operators (i.e. linear maps) between these spaces. It will also provide an introduction to spectral theory and introduce numerous examples of bounded linear operators. These objects arise naturally in applications including wavelets, signal processing and quantum mechanics, and underpin the theory of partial differential equations. The course will also introduce the Lebesgue integral on R^n and the resulting function spaces that provide the most important examples of Hilbert and Banach spaces from the point of view of applications.

#### Intended Learning Outcomes of Course

By the end of this course students will be able to:

(a) Define the basic concepts of measure theory in ℝ^n, and more general measure spaces;

(b) Define the concept of a measurable function and verify that suitable functions are measurable;

(c) Define the Lebesgue integral of a real-valued function, compute the integral of appropriate functions;

(d) State and prove the major convergence theorems (monotone convergence theorem, Fatou's lemma and the dominated convergence theorem) and use these results to establish integrability of functions and compute suitable integrals;

(e) Define the Lp spaces of subsets of ℝ^n and more general measure spaces with respect to Lebesgue measure and determine which functions lie in Lp;

(f) Define the notion of a norm and use the basic theory of normed linear spaces and operators acting on these spaces to solve simple problems;

(g) Determine whether a function on a vector space defines a norm, establish whether such norms are complete and determine when norms are equivalent;

(h) Prove the Minkowski inequality and show that the spaces ℓp and Lp are complete;

(i) Prove that all norms on a finite dimensional vector space are equivalent;

(j) State and prove equivalent formulations of boundedness for operators between normed spaces;

(k) Define the operator norm, determine whether simple operators between normed spaces are bounded and compute the norm of these operators,prove that the bounded operators between Banach spaces form a Banach space. Solve simple problems involving the operator norm;

(l) Define the concepts of linear functionals and dual spaces, use Holder's inequality to exhibit the duality between Lp and Lq (and ℓp and ℓq) for 1/p+1/q=1, show that ℓ1 is the dual space of c0 and ℓ∞ is the dual of ℓ1;

(m) State and prove the Hahn-Banach theorem and apply it to solve appropriate problems;

(n) Decide whether a given subspace of a Banach space is closed and whether it has a complement;

(o) Determine whether a bilinear form on a vector space gives an inner product, develop and use the basic theory of Hilbert spaces and orthogonality, use the Gram-Schmidt process to produce orthonormal sequences and use these in problems, describe and use properties of bases in Hilbert spaces;

(p) State and prove the Riesz representation theorem for Hilbert spaces and use this to define and establish properties of the adjoint operator, compute the adjoint of suitable operators;

(q) Define what it means for an operator to be compact, show that the compact operators are an ideal and determine whether certain operators are compact. Prove and apply the spectral theorem for compact self-adjoint operators and discuss how this theorem generalises results for finite dimensional matrices;

(r) Determine whether a densely defined unbounded symmetric operator on a Hilbert space is essentially self-adjoint or self-adjoint.

(s) Be familiar with basic examples of complete and incomplete normed linear spaces.

(t) Be familiar with shift and multiplication operators on the Hilbert space ℓ2

#### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.