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Module MA3421: Functional Analysis I

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2012-13
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
Prof. John Stalker
Learning Outcomes
On successful completion of this module, students will be able to:
  • Demonstrate a familiarity with the definitions and elementary properties of metric, normed, inner product, Banach and Hilbert spaces. This includes, but is not limited to, a knowledge of basic inequalities and identities, e.g. Cauchy-Schwarz, the Parallelogram Identity, or Bessel's Identity and Inequality;
  • Demonstrate a familiartiy with important examples of normed spaces, including Banach spaces. These should include $\mathbf R ^ n$, $l ^ p$, $L ^ p$, $C ( [ a , b ] )$. Students should also know basic theorems about linear transformations between normed spaces, such as the equivalence of boundedness and continuity. Students should not merely know the statements, and in some cases the proofs of these theorems, but should be able to apply them to concrete examples;
  • Demonstrate a knowledge and understanding of the basic theorems on Hilbert spaces, e.g. existence of orthonormal bases or the Riesz representation theorem. Students should not merely know the statements, and in some cases the proofs of these theorems, but should be able to apply them to concrete examples;
  • Demonstrate a familiarity with important classes of bounded linear transformations on a Hilbert space, such as symmetri (Hermitian) and compact operators. This includes the ability to determine whether a give linear transformation belongs to one of these classes. Give the statement of, and be able to apply, the spectral theorem for compact symmetric operators;
Module Content
  • Metric Spaces: Definitions, completeness;
  • Normed and Banach spaces: Definitions, examples, linear transformations, separability, norms of linear transformations, Riesz Lemma;
  • Inner product and Hilbert spaces: Definitions, examples, projection, orthonormal bases, Bessel's Identity and Inequality, weak versus norm convergence of sequences of vectors, weak, strong and norm convergence of sequences of operators, the Riesz Representation theorem, bilinear forms;
  • Spectral Theory: the case of compact symmetric operators;
Module Prerequisite
MA2223-Metric Spaces and MA2224-Lebesque Integral are desirable
Assessment Detail
This module will be examined jointly with MA3422 in a 3-hour examination in Trinity term, except that those taking just one of the two modules will have a 2 hour examination. There will be separate results for MA3421 and MA3422. Continuous assessment will contribute 15% to the final grade for the module at the annual examination session.