We introduce inner product spaces. After proving the Cauchy-Schwarz inequality we show that any inner product induces a norm and that the norm then satisfies the parallelogram identity. We show that the inner product can be recovered from the induced norm (via the polarisation identity). We define Hilbert spaces as complete inner product spaces and show that the spaces l^2 and L^2 are Hilbert spaces.

The Hahn-Banach Theorem allows for the extension of linear maps defined on subspaces of normed spaces to the whole space in a way that respects sublinear bounds. The simplest case is the extension of bounded linear functionals from subspaces to the whole space. We prove this result here, first for real spaces and then for complex spaces. The proof requires use of Zorn’s Lemma, unless we assume that the underlying space is separable.

We give some important applications of the Hahn-Banach Theorem. We prove the existence of a support functional and hence that X^* separates points in X. Then we prove the existence of a functional that encodes the distance from a linear subspace, which is an important ingredient in a number of subsequent proofs. We show that separability of X^* implies separability of X, define the Banach adjoint of a linear map (between Banach spaces), and prove the existence of ‘generalised Banach limits’.

We define the notion of a norm and a normed space. We prove that various canonical definitions are indeed norms (e.g. the l^p norm, the L^p norm, and the supremum norm). We discuss convergence, equivalent norms, and various notions of isomorphism between normed spaces. Finally, we discuss separability in more detail.

We discuss how to define a basis for a general normed space (a ‘Schauder basis‘). We then consider orthonormal sets in inner-product spaces and orthonormal bases for separable Hilbert spaces. We give a number of conditions that ensure that a particular orthonormal sequence forms an orthonormal basis, and as an example, we discuss the L^2 convergence of Fourier series.

We consider linear maps between normed spaces. We define what it means for a linear map to be bounded and show that this is equivalent to continuity. We define the norm of a linear operator and show that the space of all linear maps from X to Y is a vector space, which is complete if Y is complete. We give a number of examples and then discuss inverses and invertibility in some detail.

We investigate finite-dimensional normed spaces. We show that in a finite-dimensional space, all norms are equivalent, and that being compact is the same as being closed and bounded. We also show that a normed space is finite-dimensional if and only if its closed unit ball is compact, using Reisz’s Lemma.

We consider the existence of closest points in convex subsets of Hilbert sapces. In particular this enables us to define the orthogonal projection onto a closed linear subspace U of a Hilbert space H, and thereby decompose any element x of H as x=u+v, where u is an element of U and v is contained in its orthogonal complement. We also discuss ‘best approximations’ of elements of H in spaces spanned by collections of elements of H.

We show that the space C^0 is not complete in the L^1 norm and use this to motivate the abstract completion of a normed space and the Lebesgue integral. We use this approach to define the family L^p of Lebesgue spaces as the completion of the space of continuous functions in the L^p norm, and we prove some properties of these spaces.

There is a canonical way to associate an element of a Banach space X with its second dual (‘double dual’) space X^{**}. If this mapping is onto, then X is said to be reflexive. We show that Hilbert spaces, l^p and L^p for 1<p<infty, are reflexive. We then prove some general properties of reflexivity, in particular that X is reflexive if and only if X^* is reflexive, and that reflexivity is inherited by subspaces.

At the opposite extreme to the results of the last chapter, we now consider unbounded operators on Hilbert spaces. We define an appropriate notion of the Hilbert adjoint in this setting and show that for unbounded self-adjoint operators, we can develop a good spectral theory.

We recall the definition of a metric sapce, along with definitions of convergence, continuity, separability, and compactness. The treatment is intentionally brisk, but proofs are included.

We define the resolvent and spectrum of a bounded linear operator and discuss the relationship between the spectrum and the ‘point spectrum’, which is the set of all eigenvalues. We prove some basic properties of the spectrum and the Spectral Mapping Theorem for polynomials.

It is one of the major results of finite-dimensional linear algebra that all the eigenvalues of real symmetric matrices are real and that the eigenvectors of distinct eigenvalues are orthogonal. In this chapter we prove similar results for compact self-adjoint operators on infinite-dimensional HIlbert space: we show that the spectrum consists entirely of real eigenvalues (except perhaps zero), that the multiplicity of every non-zero eigenvalue is finite, and that the eigenvectors form an orthonormal basis for H. The last fact follows from the Hilbert-Schmidt Theorem, which allows us to write such operators in terms of their eigenvalues and eigenvectors.

We prove some key results about spaces of continuous functions. First we show that continuous functions on an interval can be uniformly approximated by polynomials (the Weierstrass Approximation Theorem), which has interesting applications to Fourier series. Then we prove the Stone-Weierestrass Theorem, which generalises this to continuous functions on compact metric spaces and other collections of approximating functions. We end with a proof of the Arzelà-Ascoli Theorem.