This module covers those topics that I think are most basic to modern real analysis at this level, and within the constraints imposed by a 20 hour course.
It starts with a quick run through basics: topology in metric spaces, elementary general topology, linear algebra emphasising the geometric aspects, integration on measure spaces. It covers basic theory of Banach spaces - complete normed linear spaces - and linear operators on them. An important example is the class of Hilbert spaces - Banach spaces with an inner product - which includes the L2
spaces, of functions square-integrable with respect to a measure.
The latter part of the course is angled to a fundamental theorem on Fourier Series (FS) of periodic functions on the real line R
, dating from the late 19th century. Namely, the FS of every L2
periodic function f converges to f in the L2
sense (whereas the corresponding result for L1
functions, and for continuous functions, is false).
Finally, the tools developed are deployed to prove another big 19th century result, the Riemann-Lebesgue Lemma: the Fourier transform of an L1
function on R
is continuous and converges to 0 at infinity.
The material is all in my 1973 book "Basic Methods of Linear Functional Analysis" (see Bibliography). It was reprinted by Dover Press in December 2011 (with many corrections-the new edition is a better product). Amazon price in Jan 2014 is £14.67 including free delivery in the UK, but other sellers can get it to you for under £9. Barnes and Noble also have it cheap, if you happen to be visiting the USA.
This summarises what I taught in 2012-13. It covers most of the syllabus I was originally given. Stuff in [square brackets] is not explicitly in that syllabus.
This time I aim to cover basic theory of the adjoint of an operator on a normed space, hence maybe omit some material at the end.
Basic metric spaces. [Topologies]. Separability. [Compactness]. [Metric space properties equivalent to compactness, e.g. Weierstrass convergent subsequence condition].
Basic measure theory, the integral, and basic theorems such as Dominated Convergence. The Lp
spaces on an arbitrary measure space. Existence and regularity of Lebesgue measure assumed without proof.
Basic linear spaces, [linearly independent and spanning sets, product and quotient of linear spaces]. [Linear maps, image, kernel.]
[Algebraic operations on subsets of linear space; geometric viewpoint; algebra of convex subsets.]
Normed linear spaces. Definitions and examples.
Basic normed space topology and geometry. [Compactness, convexity in normed spaces].
Completeness and how to establish it. Characterisation by infinite series.
Proof that L1
is complete. (Outline proof) general Lp
is complete. Lp
with Lebesgue measure, on a real interval, is separable for 1 < = p < ∞.
[Subspace, product, quotient of normed spaces].
Linear maps and functionals on normed spaces. [Operator norm. Space Bdd(X,Y) of bounded operators. Bdd(X,Y) is a Banach space if Y is.]
The dual space of a normed space. [Hahn-Banach Theorem mentioned; proved later].
Inner product spaces. Hilbert spaces. Examples, including L2
Schwarz inequality, generalised Pythagoras, [polarisation identities].
Orthogonality, [orthogonal complement, projection].
Riesz-Fréchet Theorem on bounded linear functionals on a Hilbert space.
Orthonormal sets and expansions. Bessel's inequality, Parseval's equation. Characterisations of complete ONS. [Characterisations of separable Hilbert spaces].
[Overview of Zorn's Lemma].
[The Fourier series convergence problem].
[Density theorems. Fundamental sets in a normed space.]
[Weierstrass trigonometric polynomial approximation theorem (Fejér's proof)].
[Hence the functions eint
are a complete orthogonal set in L2
[Bounded operator sequence theorem. Application to Riemann-Lebesgue Lemma and related topics.]
[Proof of Hahn-Banach Theorem.]
A few items in the syllabus I was given are only implicitly present in the above: e.g. Lusin's Theorem, which is closely related to the regularity of Lebesgue measure.