文档介绍：Dedicated to the memory of my parents:
Douglas McDonald Bridges and Allison Hogg
Sweet Analytics, ’tis thou hast ravished me.
Faustus (Marlowe)
The stone which the builders refused is e the head stone
of the corner.
Psalm cxviii, 22.
...from so simple a beginning endless forms most beautiful and
most wonderful have been, and are being, evolved.
The origin of species (Darwin)
Preface
The core of this book, Chapters 3 through 5, presents a course on metric,
normed, and Hilbert spaces at the senior/graduate level. The motivation for
each of these chapters is the generalisation of a particular attribute of the
Euclidean space Rn: in Chapter 3, that attribute is distance; in Chapter 4,
length; and in Chapter 5, inner product. In addition to the standard topics
that, arguably, should form part of the armoury of any graduate student
in mathematics, physics, mathematical economics, theoretical statistics,...,
this part of the book contains many results and exercises that are seldom
found in texts on analysis at this level. Examples of the latter are Wong’s
Theorem () showing that the Lebesgue covering property is equivalent
to the uniform continuity property, and Motzkin’s result () that a
nonempty closed subset of Euclidean space has the unique closest point
property if and only if it is convex.
The sad reality today is that, perceiving them as one of the harder parts
of their mathematical studies, students contrive to avoid analysis courses at
almost any cost, in particular that of their own educational and technical
deprivation. Many universities have at times capitulated to the negative
demand of students for analysis courses and have seriously watered down
their expectations of students in that area. As a result, mathematics ma-
jors are graduating, sometimes with high honours, with little exposure to
anything but a rudimentary course or two on real plex analysis,
often without even an introduction to the Lebesgue integral