Single-Course English 5 ECTS

# Functional Analysis

## Overall Course Objectives

Many advanced mathematical disciplines, e.g., Fourier analysis, numerical analysis, the calculus of variations, the theory of differential and integral operators, have a common foundation incorporating among others fundamental notions and abstractions associated with linear spaces of functions (functional analysis). It is the purpose of this course to provide students with basic concepts from functional analysis to facilitate the study of advanced mathematical structures arising in the natural sciences and the engineering sciences and to grasp the newest technical and mathematical literature.

Learning Objectives

• List function spaces and sequence spaces used in functional analysis, and list their main properties.
• State and relate the definitions of metric spaces, normed vector space, Banach spaces, dual spaces, reflexive spaces, inner product spaces, and Hilbert spaces.
• Explain fundamental concepts as isomorphisms, completeness, separability, orthogonality, and dimension.
• State, relate, and determine different types of convergence in normed vector spaces.
• Construct mathematical proofs of a methodical character.
• State, explain, and prove the fundamental theorems in functional analysis.
• Identify and recast concrete problems in theoretical parts of the natural and engineering sciences in terms of abstract functional analytic notions, such as Banach spaces and Hilbert spaces.
• Solve concrete mathematical problems by abstract results from functional analysis

Course Content

Bounded and unbounded linear operators in normed vector spaces. Completion of normed vector spaces. Construction of L^p-spaces by completion of spaces of continuous functions. General Banach space and Hilbert space theory. The projection theorem. Bounded and unbounded operators on Hilbert spaces. Weak, weak-*, and strong convergence. Baire’s category theorem. The Hahn-Banach theorem. The Open Mapping Theorem. The Uniform Boundedness theorem. The Closed Graph theorem. The Spectral Theorem for compact, self-adjoint operators on separable Hilbert spaces. Applications of functional analytical methods within, e.g., PDEs, wavelet theory, frame theory, or optimization.

Recommended prerequisites

01325/01125, It will be very difficult to complete the course without having already passed both “01125 Fundamental topological
concepts and metric spaces” and ” 01325 Function spaces and mathematical analysis”

Teaching Method

Lectures, where fundamental concepts, methods, and results are presented and put into perspective, and tutorials, where the theory is exemplified by the solution of exercises.

Faculty

See course in the course database.

## Registration

Language English 13 weeks Compute DTU Lyngby Campus 01715 Candidate Week 35 Week 48 Fri 8-12
Price 7.500,00 DKK