Finite-Dimensional Normed Spaces

In this course we shall study the classical theory of Banach spaces with an eye to its quantitative aspects. The overarching structure follows that of the notes [Gar03] by Garling entitled ‘Classical Banach Spaces’, but we also borrow heavily from the notes [Nao10] of Naor entitled ‘Local Theory of Banach Spaces’, and the book [Woj91] of Wojtaszczyk entitled ‘Banach Spaces for Analysts’. In terms of prerequisites it will be useful to have taken a basic course on Banach spaces. In the Oxford undergraduate degree there are three particularly helpful courses: (a) B4.1 Banach Spaces, maths.ox.ac.uk/courses/course/26298/synopsis; (b) B4.2 Hilbert Spaces, maths.ox.ac.uk/courses/course/26299/synopsis; (c) C4.1 Functional Analysis, maths.ox.ac.uk/courses/course/26335/synopsis. To agree notation we shall recap the relevant material when we come to need it, and while we shall not dwell on ideas already developed in other courses we shall try to direct the interested reader to a suitable source. Finally, the book [Bol99] of Bollobás may also serve as a useful companion. The course is constructed from the perspective that examples are essential, and there will be an examples sheet available at people.maths.ox.ac.uk/sanders/ to which problems will be added.