Applied Functional Analysis (MAGIC062) |
GeneralThis course is part of the MAGIC core. Description
I feel a better name for this module is 'Introductory Functional Analysis'. It 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.
Starting with topology in metric 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 L_{2} 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 L_{2} periodic function f converges to f in the L_{2} sense (whereas the corresponding result for L_{1} 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 L_{1} 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 has been reprinted by Dover Press and is available in UK since December 2011. Amazon price it at GBP17.99 including free delivery in the UK, though Barnes and Noble are quoting the far cheaper price of USD13.43, if you happen to be visiting the States.
SemesterSpring 2012 (Monday, January 16 to Friday, March 23) Timetable
PrerequisitesStandard undergraduate real analysis.
SyllabusNote: I regard this as more 'Introductory' Functional Analysis than 'Applied'.
This summarises what I taught in 2011. It covers most of the syllabus I was originally given. Stuff in [square brackets] is not explicitly in that syllabus.
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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 L_{p} 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 L_{1} is complete. (Outline proof) general L_{p} is complete. L_{p} 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 briefly].
Inner product spaces. Hilbert spaces. Examples, including L_{2} spaces.
Schwarz inequality, Pythagoras, [polarisation identities].
Orthogonality, [orthogonal complement, projection].
Riesz-Fr'echet 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 e^{int} are a complete orthogonal set in L_{2}[−pi,pi]].
[Bounded operator sequence theorem. Application to Riemann-Lebesgue Lemma and related topics.]
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.
Lecturer
Students
Bibliography
Note: Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.) Assessment(28 March 2012)
Exercises 2 is now posted and I have changed its deadline. It is now after the Easter break, Friday 27 April.
My provisional mark scheme, reflecting my estimate of the difficulty of each question, is: Q1=10, Q2=13, Q3=14, Q4=13.
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There will be two assessed exercises, each out of 50, deadlines 16 March and 6 April 2012. Do all questions.
I haven't finalised the mark scheme, but clearly Qs 2 and 6 are more substantial than the others and will carry about twice the mark: say 12 each, versus 6 or 7 each for the others.
Exercises on blocks 1 to 4 of the course
Exercises on later material of the course
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