Mathematical Institute

Metric Spaces (10 lectures)

Basic definitions: metric spaces, isometries, continuous functions (\(\varepsilon-\delta\) definition), homeomorphisms, open sets, closed sets. Examples of metric spaces, including metrics derived from a norm on a real vector space, particularly \(l^1, l^2, l^\infty\) norms on \(\mathbb{R}^n\), the sup norm on the bounded real-valued functions on a set, and on the bounded continuous real-valued functions on a metric space. The characterisation of continuity in terms of the pre-image of open sets or closed sets. The limit of a sequence of points in a metric space. A subset of a metric space inherits a metric. Discussion of open and closed sets in subspaces. The closure of a subset of a metric space. [3]

Completeness (but not completion). Completeness of the space of bounded real-valued functions on a set, equipped with the norm, and the completeness of the space of bounded continuous real-valued functions on a metric space, equipped with the metric. Lipschitz maps and contractions. Contraction Mapping Theorem. [2.5]

Connected metric spaces, path-connectedness. Closure of a connected space is connected, union of connected sets is connected if there is a non-empty intersection, continuous image of a connected space is connected. Path-connectedness implies connectedness. Connected open subset of a normed vector space is path-connected. [2]

Definition of sequential compactness and proof of basic properties of compact sets. Preservation of compactness under continuous maps, equivalence of continuity and uniform continuity for functions on a compact set. Equivalence of sequential compactness with being complete and totally bounded. The Arzela-Ascoli theorem (proof non-examinable). Open cover definition of compactness. Heine-Borel (for the interval only) and proof that compactness implies sequential compactness (statement of the converse only). [2.5]

Complex Analysis (22 lectures)

Basic geometry and topology of the complex plane, including the equations of lines and circles. Extended complex plane, Riemann sphere, stereographic projection. Möbius transformations acting on the extended complex plane. Möbius transformations take circlines to circlines. [3]

Complex differentiation. Holomorphic functions. Cauchy-Riemann equations (including \(z,\bar{z}\) version). Real and imaginary parts of a holomorphic function are harmonic. [2]

Recap on power series and differentiation of power series. Exponential function and logarithm function. Fractional powers — examples of multifunctions. The use of cuts as method of defining a branch of a multifunction. [3]

Path integration. Cauchy's Theorem. (Sketch of proof only — students referred to various texts for proof.) Fundamental Theorem of Calculus in the path integral/holomorphic situation. [2]

Cauchy's Integral formulae. Taylor expansion. Liouville's Theorem. Identity Theorem. Morera's Theorem. [4]

Laurent's expansion. Classification of isolated singularities. Calculation of principal parts, particularly residues. [2]

Residue Theorem. Evaluation of integrals by the method of residues (straightforward examples only but to include the use of Jordan's Lemma and simple poles on contour of integration). [3]

Conformal mappings. Riemann mapping theorem (no proof), Möbius transformations, exponential functions, fractional powers; mapping regions (not Christoffel transformations or Joukowski's transformation). [3]

Brief recall of material from Part A Metric Spaces and Part A Linear Algebra on real and complex normed vector spaces, their geometry and topology and simple examples of completeness. The norm associated with an inner product and its properties. Banach spaces, exemplified by \(\ell^p\), \(L^p\), \(C(K)\), spaces of differentiable functions. Finite-dimensional normed spaces, including equivalence of norms and completeness. Hilbert spaces as a class of Banach spaces having special properties (illustrations, but no proofs); examples (Euclidean spaces, \(\ell^2\), \(L^2\)).

Density. Approximation of functions, Stone-Weierstrass Theorem. Separable spaces; separability of subspaces.

Bounded linear operators, examples (including integral operators). Continuous linear functionals. Dual spaces. Hahn-Banach Theorem (proof for separable spaces only); applications, including density of subspaces and embedding of a normed space into its second dual. Adjoint operators.

Spectrum and resolvent. Spectral mapping theorem for polynomials.

The Fourier transform on \(\mathbb{R}^n\): the Schwartz class \(\mathcal{S}\) of test functions on \(\mathbb{R}^n\) , properties of the Fourier transform on \(\mathcal{S}\), the Fourier transform of a Gaussian and the inversion formula on \(\mathcal{S}\). [4 lectures]

The class of tempered distributions \(\mathcal{S}'\) and their calculus. Fourier transforms of tempered distributions: definitions and examples, convolutions with tempered distributions. The inversion formula on \(\mathcal{S}'\). Fourier transfor in \(L^2\) and ~Plancherel's theorem. [5 lectures]

Solving PDEs using Fourier transformation: the Laplace equation, the heat equation, the wave equation, Schrödinger's equation. Fundamental solutions, ellipticity and hypoellipticity. [3 lectures]

Fourier Analysis: the Riemann-Lebesgue lemma, Paley-Wiener theorems, the Poisson summation formula, the uncertainty principle.[4 lectures]

We will cover the following topics:

• The Riemann mapping theorem. The main goal will be to prove Riemann's theorem which tells us that any non-trivial simply-connected domain can be conformally mapped onto the unit disc. This will be the key result for the entire course since it will allow us to connect the geometry of a domain with the analytical properties of a conformal map which sends this domain to the unit disc. Within this section we will discuss

1. Möbius transformations and the Schwarz lemma
2. Normal families, the Montel and Hurwitz theorems
3. Proof of Riemann mapping theorem
4. Constructive uniformization: Christoffel-Schwarz mappings and iterative methods
5. Boundary correspondence, accessible points and prime ends
6. Uniformization of multiply-connected domains (without proof)
7. Applications: the Dirichlet boundary problem

1. Area theorem and coefficient estimates
2. The Koebe \(1/4\) theorem, distortion theorems
3. Conformal invariants: extremal length and its applications

1. Conformal invariance of Green's function and harmonic measure
2. Extremal length, its conformal invariance and applications

Definition of the function limit. Definition of continuity of functions on subsets of \(\mathbb{R}\) and \(\mathbb{C}\) in terms of \(\varepsilon\) and \(\delta\). Continuity of real valued functions of several variables. The algebra of continuous functions; examples, including polynomials. Intermediate Value Theorem for continuous functions on intervals. Boundedness, maxima, minima and uniform continuity for continuous functions on closed intervals. Monotone functions on intervals and the Inverse Function Theorem.

Sequences and series of functions, uniform convergence. Weierstrass's M-test for uniformly convergent series of functions. Uniform limit of a sequence of continuous functions is continuous. Continuity of functions defined by power series.

Definition of the derivative of a function of a real variable. Algebra of derivatives, examples to include polynomials and inverse functions. The derivative of a function defined by a power series is given by the derived series (proof not examinable). Vanishing of the derivative at a local maximum or minimum. Rolle's Theorem, Mean Value Theorem, and Cauchy's (Generalized) Mean Value Theorem with applications: Constancy Theorem, monotone functions, exponential function and trigonometric functions. L'Hôpital's Formula. Taylor's Theorem with remainder in Lagrange's form; examples. The binomial expansion with arbitrary index.

Real numbers: arithmetic, ordering, suprema, infima; the real numbers as a complete ordered field. Definition of a countable set. The countability of the rational numbers. The reals are uncountable. The complex number system. The triangle inequality.

Sequences of real or complex numbers. Definition of a limit of a sequence of numbers. Limits and inequalities. The algebra of limits. Order notation: \(O\), \(o\).

Subsequences; a proof that every subsequence of a convergent sequence converges to the same limit; bounded monotone sequences converge. Bolzano--Weierstrass Theorem. Cauchy's convergence criterion.

Series of real or complex numbers. Convergence of series. Simple examples to include geometric progressions and some power series. Absolute convergence, Comparison Test, Ratio Test, Integral Test. Alternating Series Test.

Power series, radius of convergence. Examples to include definition of and relationships between exponential, trigonometric functions and hyperbolic functions.

Metric Spaces (10 lectures)

Basic definitions: metric spaces, isometries, continuous functions (\(\varepsilon-\delta\) definition), homeomorphisms, open sets, closed sets. Examples of metric spaces, including metrics derived from a norm on a real vector space, particularly \(l^1, l^2, l^\infty\) norms on \(\mathbb{R}^n\), the sup norm on the bounded real-valued functions on a set, and on the bounded continuous real-valued functions on a metric space. The characterisation of continuity in terms of the pre-image of open sets or closed sets. The limit of a sequence of points in a metric space. A subset of a metric space inherits a metric. Discussion of open and closed sets in subspaces. The closure of a subset of a metric space. [3]

Completeness (but not completion). Completeness of the space of bounded real-valued functions on a set, equipped with the norm, and the completeness of the space of bounded continuous real-valued functions on a metric space, equipped with the metric. Lipschitz maps and contractions. Contraction Mapping Theorem. [2.5]

Connected metric spaces, path-connectedness. Closure of a connected space is connected, union of connected sets is connected if there is a non-empty intersection, continuous image of a connected space is connected. Path-connectedness implies connectedness. Connected open subset of a normed vector space is path-connected. [2]

Definition of sequential compactness and proof of basic properties of compact sets. Preservation of compactness under continuous maps, equivalence of continuity and uniform continuity for functions on a compact set. Equivalence of sequential compactness with being complete and totally bounded. The Arzela-Ascoli theorem (proof non-examinable). Open cover definition of compactness. Heine-Borel (for the interval only) and proof that compactness implies sequential compactness (statement of the converse only). [2.5]

Complex Analysis (22 lectures)

Basic geometry and topology of the complex plane, including the equations of lines and circles. Extended complex plane, Riemann sphere, stereographic projection. Möbius transformations acting on the extended complex plane. Möbius transformations take circlines to circlines. [3]

Complex differentiation. Holomorphic functions. Cauchy-Riemann equations (including \(z,\bar{z}\) version). Real and imaginary parts of a holomorphic function are harmonic. [2]

Recap on power series and differentiation of power series. Exponential function and logarithm function. Fractional powers — examples of multifunctions. The use of cuts as method of defining a branch of a multifunction. [3]

Path integration. Cauchy's Theorem. (Sketch of proof only — students referred to various texts for proof.) Fundamental Theorem of Calculus in the path integral/holomorphic situation. [2]

Cauchy's Integral formulae. Taylor expansion. Liouville's Theorem. Identity Theorem. Morera's Theorem. [4]

Laurent's expansion. Classification of isolated singularities. Calculation of principal parts, particularly residues. [2]

Residue Theorem. Evaluation of integrals by the method of residues (straightforward examples only but to include the use of Jordan's Lemma and simple poles on contour of integration). [3]

Conformal mappings. Riemann mapping theorem (no proof): Möbius transformations, exponential functions, fractional powers; mapping regions (not Christoffel transformations or Joukowski's transformation). [3]

Brief recall of material from Part A Metric Spaces and Part A Linear Algebra on real and complex normed vector spaces, their geometry and topology and simple examples of completeness. The norm associated with an inner product and its properties. Banach spaces, exemplified by \(\ell^p\), \(L^p\), \(C(K)\), spaces of differentiable functions. Finite-dimensional normed spaces, including equivalence of norms and completeness. Hilbert spaces as a class of Banach spaces having special properties (illustrations, but no proofs); examples (Euclidean spaces, \(\ell^2\), \(L^2\)).

Density. Approximation of functions, Stone-Weierstrass Theorem. Separable spaces; separability of subspaces.

Bounded linear operators, examples (including integral operators). Continuous linear functionals. Dual spaces. Hahn-Banach Theorem (proof for separable spaces only); applications, including density of subspaces and embedding of a normed space into its second dual. Adjoint operators.

Spectrum and resolvent. Spectral mapping theorem for polynomials.

Hilbert spaces; examples including \(L^2\)-spaces. Orthogonality, orthogonal complement, closed subspaces, projection theorem. Riesz Representation Theorem.

Linear operators on Hilbert space, adjoint operators. Self-adjoint operators, orthogonal projections, unitary operators.

Baire Category Theorem and its consequences for operators on Banach spaces (Uniform Boundedness, Open Mapping, Inverse Mapping and Closed Graph Theorems). Strong convergence of sequences of operators.

Weak convergence. Weak precompactness of the unit ball.

Spectral theory in Hilbert spaces, in particular spectra of self-adjoint and unitary operators.

Orthonormal sets, Pythagoras, Bessel's inequality. Complete orthonormal sets, Parseval. \(L^2\)-theory of Fourier series, including completeness of the trigonometric system. Examples of other orthogonal expansions (Legendre, Laguerre, Hermite etc.).

Brief contextual comments on the classical theory of Fourier series and modes of convergence; exposition of failure of pointwise convergence of Fourier series of some continuous functions.

Normed vector spaces and Banach spaces. Dual spaces. Direct sums and complemented subspaces. Quotient spaces and quotient operators.

The Baire Category Theorem and its consequences (review).

Hahn-Banach extension and separation theorems. The bidual space. Reflexivity. Completion of a normed vector space.

Convexity and smoothness of norms. Lebesgue spaces and their duals.

Weak and weak* topologies. The Banach-Alaoglu theorem. Goldstine's theorem. Equivalence of reflexivity and weak compactness of the closed unit ball. The Schur property of \(\ell^1\). Weakly compact operators.

Compactness in normed vector spaces. The Arzelà-Ascoli theorem. Compact operators. Schauder's theorem on compactness of dual operators. Completely continuous operators.

The Closed Range Theorem. Fredholm theory: Fredholm operators; the Fredholm index; perturbation results; the Fredholm Alternative. Spectral theory of compact operators. The Spectral Theorem for compact self-adjoint operators.

Schauder bases; examples in classical Banach spaces.