# B4.2 Functional Analysis II - Material for the year 2020-2021

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B4.1 Functional Analysis I is an essential pre-requisite. A4 Integration is also 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.

16 lectures

### Assessment type:

- Written Examination

The course provides further introduction to the methods of functional analysis. It builds on core material in Part A analysis and linear algebra and in Part B B4.1 Functional Analysis I. On one hand, it delves deeper into operator theory on Banach spaces, and on the other, it gives a systematic study of Hilbert spaces, operators on Hilbert spaces and their special features. The techniques and examples studied in the course, together with that in B4.1, 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.''

Students will appreciate the role of completeness through the Baire category theorem and its consequences for operators on Banach spaces. They will have a demonstrable knowledge of the properties of a Hilbert space, including orthogonal complements, orthonormal sets, complete orthonormal sets together with related identities and inequalities. They will be familiar with the theory of linear operators on a Hilbert space, including adjoint operators, self-adjoint and unitary operators with their spectra. They will know the $L^2$-theory of Fourier series and be aware of the classical theory of Fourier series and other orthogonal expansions.

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.

- B.P. Rynne and M.A. Youngson,
*Linear Functional Analysis*(Springer SUMS, 2nd edition, 2008), Chapters 3, 4.4, 6. - E. Kreyszig,
*Introductory Functional Analysis with Applications*(Wiley, revised edition, 1989), Chapters 3, 4.7-4.9, 4.12-4.13, 9.1-9.2. - N. Young,
*An Introduction to Hilbert Space*(Cambridge University Press, 1988), Chs 1-7.

- E.M. Stein and R. Shakarchi,
*Real Analysis: Measure Theory, Integration & Hilbert Spaces*(Princeton Lectures in Analysis III, 2005), Chapter 4. - M. Reed and B. Simon,
*Methods of Modern Mathematical Physics Vol. I. Functional Analysis*(Academic Press, 1980). - H. Brezis,
*Functional Analysis, Sobolev Spaces and Partial Differential Equations*(Universitext. Springer, 2011). - P. Lax,
*Functional Analysis*(Wiley, 2012).