We introduce the property of completeness and prove some abstract results about complete normed spaces (Banach spaces). We then give a number of examples of Banach spaces: l^p, L^p, and spaces of continuous functions. We then discuss convergence of series in Banach spaces and end the chapter with a proof of the Contraction Mapping Theorem.

Using the Baire Category Theorem, we prove the Principle of Uniform Boundedness, which allows us to deduce uniform bounds on collections of bounded linear operators from pointwise properties. We use the powerful corollary known as the “Condensation of Singularities” to show that there are continuous periodic functions whose Fourier series do not converge pointwise everywhere.

We define weak convergence and give some examples, including a proof of Schur’s Theorem that weak and strong convergence coincide in l^1. We also show that closed convex subsets of Banach spaces are weakly closed. We then introduce weak-* convergence and prove two powerful weak compactness theorems: Helly’s Theorem for weak-* convergence in the duals of separable Banach spaces and a weak sequential compactness theorem in reflexive Banach spaces.

We define weak convergence and give some examples, including a proof of Schur’s Theorem that weak and strong convergence coincide in l^1. We also show that closed convex subsets of Banach spaces are weakly closed. We then introduce weak-* convergence and prove two powerful weak compactness theorems: Helly’s Theorem for weak-* convergence in the duals of separable Banach spaces and a weak sequential compactness theorem in reflexive Banach spaces.

As a corollary of the Hahn-Banach Theorem, we show that any two convex sets can be separated using a linear functional; a key ingredient is the definition of the Minkowski functional of a convex set. This separation theorem allows us to give a characterisation of convex sets in terms of their supporting hyperplanes that will be useful later. We then define the closed convex hull of a set, introduce the notion of extreme points in a convex set, and prove the Krein-Milman Theorem: a non-empty compact convex subset of a Banach space is the closed convex hull of its extreme points.

We define weak convergence and give some examples, including a proof of Schur’s Theorem that weak and strong convergence coincide in l^1. We also show that closed convex subsets of Banach spaces are weakly closed. We then introduce weak-* convergence and prove two powerful weak compactness theorems: Helly’s Theorem for weak-* convergence in the duals of separable Banach spaces and a weak sequential compactness theorem in reflexive Banach spaces.

Once again we use the Baire Category Theorem to prove results about linear maps between Banach spaces. We prove the Open Mapping Theorem and, as a corollary, the Inverse Mapping Theorem, which allows for some simplification in the spectral theory of bounded operators. As an application, we prove the existence of a ‘basis constant’ for any Schauder basis in a separable Banach space. Finally, we use the Inverse Mapping Theorem to prove the Closed Graph Theorem, which gives an alternative way to check whether a linear map T from X into Y is bounded, provided both X and Y are Banach spaces.

We return to the topic of dual spaces. We prove Young’s inequality and Hölder’s inequality in the l^p and L^p spaces. We identify the dual spaces of l^p and L^p up to isometric isomorphisms: the proofs for l^p are presented in full, with the measure-theoretic proofs for L^p contained in Appendix B.

We introduce the dual space of a Banach space X, which is the collection of all bounded linear maps from X into the field K (‘linear functionals’), equipped with the corresponding operator norm. We prove the Riesz Representation Theorem, which shows that in a Hilbert space, any linear functional can be written as the inner product with some element of the Hilbert space.

We define what it means for a linear operator to be compact and show that the space of all compact linear operators is complete (with the operator norm). We give some examples and show that the spectrum of a compact operator from an infinite-dimensional space into itself is always non-empty (it must contain zero).

We recall some of the basic theory of linear algebra, beginning with the formal definition of a vector space. We then discuss linear maps between vector spaces and end by proving that every vector space has a basis using Zorn’s Lemma.

We define weak convergence and give some examples, including a proof of Schur’s Theorem that weak and strong convergence coincide in l^1. We also show that closed convex subsets of Banach spaces are weakly closed. We then introduce weak-* convergence and prove two powerful weak compactness theorems: Helly’s Theorem for weak-* convergence in the duals of separable Banach spaces and a weak sequential compactness theorem in reflexive Banach spaces.