General Prerequisites: Part A Integration is essential; the only concepts which will be used are the convergence theorems and the theorems of Fubini and Tonelli, and the notions of measurable functions, integrable functions, null sets and \(L^p\) spaces. No knowledge is needed of outer measure, or of any particular construction of the integral, or of any proofs. A good working knowledge of Part A Core Analysis (both metric spaces and complex analysis) is expected.

Course Overview: The course provides an introduction to the methods of functional analysis.

It builds on core material in analysis and linear algebra studied in Part A. The focus is on normed spaces and Banach spaces; a brief introduction to Hilbert spaces is included, but a systematic study of such spaces and their special features is deferred to B4.2 Functional Analysis II. The techniques and examples studied in the Part B courses Functional Analysis I and II support, in a variety of ways, many advanced courses, in particular in analysis and partial differential equations, as well as having applications in mathematical physics and other areas.

Learning Outcomes: Students will have a firm knowledge of real and complex normed vector spaces, with their geometric and topological properties. They will be familiar with the notions of completeness, separability and density, will know the properties of a Banach space and important examples, and will be able to prove results relating to the Hahn-Banach Theorem. They will have developed an understanding of the theory of bounded linear operators on a Banach space.

Course Synopsis: 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.