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Unit of study_

MATH4313: Functional Analysis

Functional analysis is one of the major areas of modern mathematics. It can be thought of as an infinite-dimensional generalisation of linear algebra and involves the study of various properties of linear continuous transformations on normed infinite-dimensional spaces. Functional analysis plays a fundamental role in the theory of differential equations, particularly partial differential equations, representation theory, and probability. In this unit you will cover topics that include normed vector spaces, completions and Banach spaces; linear operators and operator norms; Hilbert spaces and the Stone-Weierstrass theorem; uniform boundedness and the open mapping theorem; dual spaces and the Hahn-Banach theorem; and spectral theory of compact self-adjoint operators. A thorough mechanistic grounding in these topics will lead to the development of your compositional skills in the formulation of solutions to multifaceted problems. By completing this unit you will become proficient in using a set of standard tools that are foundational in modern mathematics and will be equipped to proceed to research projects in PDEs, applied dynamics, representation theory, probability, and ergodic theory.

Code MATH4313
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
Assumed knowledge:
Real Analysis and abstract linear algebra (e.g., MATH2X23 and MATH2X22 or equivalent), and, preferably, knowledge of Metric Spaces

At the completion of this unit, you should be able to:

  • LO1. demonstrate a coherent and advanced understanding of the key concepts of geometry of normed spaces, Hilbert Space Theory, Abstract Fourier Analysis, Hahn-Banach Theory and Spectral Theory, and how they provide a unified approach to infinite-dimensional linear problems in mathematics
  • LO2. apply the fundamental ideas and results in functional analysis to solve given problems
  • LO3. distinguish and compare the properties of different types of linear operators, analysing their spectra and deriving their main properties
  • LO4. formulate analytic problems in functional-analytic terms and determine the appropriate framework to solve them
  • LO5. communicate coherent mathematical arguments appropriately to student and expert audiences, both orally and through written work
  • LO6. devise computational solutions to complex problems in functional analysis
  • LO7. compose correct proofs of unfamiliar general results in functional analysis.