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Functional Analysis, 6 credits
Funktionalanalys, 6 hp
TATM85
Main field of study
Mathematics Applied MathematicsCourse level
Second cycleCourse type
Programme courseExaminer
Bengt Ove TuressonDirector of studies or equivalent
Jesper ThorénEducation components
Preliminary scheduled hours: 60 hRecommended self-study hours: 100 h
Available for exchange students
YesECV = Elective / Compulsory / Voluntary
Main field of study
Mathematics, Applied MathematicsCourse level
Second cycleAdvancement level
A1XCourse offered for
- Mathematics, Master's Programme
- Applied Physics and Electrical Engineering - International, M Sc in Engineering
- Applied Physics and Electrical Engineering, M Sc in Engineering
Entry requirements
Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.
Prerequisites
Linear Algebra,Calculus in one variable, Calculus in several variables, Real Analysis Honours Course, is a good preparation, but is not a prerequisite.Intended learning outcomes
To give familiarity with the fundamental concepts of functional analysis and to prepare the student for further studies in mathematics and related areas. After completing this course the student is able to:
- Describe the concepts of a metric space, vector space, subspace, normed space, Banach space, Hilbert space and dual space.
- Describe the logical connections between important concepts in the spaces mentioned above.
- Describe and apply central theorems in the theory of metric spaces, normed spaces, Banach spaces and Hilbert spaces.
- Apply Banach's fix point theorem in the theory of differential and integral equations.
- Describe the spectral theory for compact self adjoint operators
- Describe the Lebesgue integral and apply central theorems in the theory of the Lebesgue integral in order to solve problems.
Course content
Elementary metric space theory. Normed spaces, Banach spaces, Hilbert spaces. Linear functionals, dual spaces. Linear operators. Aspects of the Lebesgue integral. Integral equations. Introduction to spectral theory for compact, symmetric operators.
Teaching and working methods
Lectures
The course runs over the entire autumn semester.
Examination
| UPG1 | Assignments | 6 credits | U, 3, 4, 5 |
Grades
Four-grade scale, LiU, U, 3, 4, 5Other information
Supplementary courses: Partial differential equations, Fourier and Wavelet Analysis
Department
Matematiska institutionenDirector of Studies or equivalent
Jesper ThorénExaminer
Bengt Ove TuressonCourse website and other links
www.mai.liu.se/und/kurser/index-amne-tm.htmlEducation components
Preliminary scheduled hours: 60 hRecommended self-study hours: 100 h
Course literature
Additional literature
Books
- Kreyszig, E, (1981) Introductory Functional Analysis with Applications
Compendia
Other
| Code | Name | Scope | Grading scale |
|---|---|---|---|
| UPG1 | Assignments | 6 credits | U, 3, 4, 5 |
Regulations (apply to LiU in its entirety)
The university is a government agency whose operations are regulated by legislation and ordinances, which include the Higher Education Act and the Higher Education Ordinance. In addition to legislation and ordinances, operations are subject to several policy documents. The Linköping University rule book collects currently valid decisions of a regulatory nature taken by the university board, the vice-chancellor and faculty/department boards.
LiU’s rule book for education at first-cycle and second-cycle levels is available at http://styrdokument.liu.se/Regelsamling/Innehall/Utbildning_pa_grund-_och_avancerad_niva.
Additional literature
Books
Kreyszig, E, (1981) Introductory Functional Analysis with Applications
Compendia
Other
Note: The course matrix might contain more information in Swedish.
I = Introduce, U = Teach, A = Utilize
| I | U | A | Modules | Comment | ||
|---|---|---|---|---|---|---|
| 1. DISCIPLINARY KNOWLEDGE AND REASONING | ||||||
| 1.1 Knowledge of underlying mathematics and science (G1X level) |
X
|
X
|
X
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UPG1
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| 1.2 Fundamental engineering knowledge (G1X level) |
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| 1.3 Further knowledge, methods, and tools in one or several subjects in engineering or natural science (G2X level) |
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| 1.4 Advanced knowledge, methods, and tools in one or several subjects in engineering or natural sciences (A1X level) |
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| 1.5 Insight into current research and development work |
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| 2. PERSONAL AND PROFESSIONAL SKILLS AND ATTRIBUTES | ||||||
| 2.1 Analytical reasoning and problem solving |
|
X
|
X
|
UPG1
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| 2.2 Experimentation, investigation, and knowledge discovery |
|
X
|
X
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| 2.3 System thinking |
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| 2.4 Attitudes, thought, and learning |
|
X
|
X
|
UPG1
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| 2.5 Ethics, equity, and other responsibilities |
|
X
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| 3. INTERPERSONAL SKILLS: TEAMWORK AND COMMUNICATION | ||||||
| 3.1 Teamwork |
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| 3.2 Communications |
|
X
|
X
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| 3.3 Communication in foreign languages |
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X
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| 4. CONCEIVING, DESIGNING, IMPLEMENTING AND OPERATING SYSTEMS IN THE ENTERPRISE, SOCIETAL AND ENVIRONMENTAL CONTEXT | ||||||
| 4.1 External, societal, and environmental context |
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| 4.2 Enterprise and business context |
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| 4.3 Conceiving, system engineering and management |
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| 4.4 Designing |
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| 4.5 Implementing |
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| 4.6 Operating |
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| 5. PLANNING, EXECUTION AND PRESENTATION OF RESEARCH DEVELOPMENT PROJECTS WITH RESPECT TO SCIENTIFIC AND SOCIETAL NEEDS AND REQUIREMENTS | ||||||
| 5.1 Societal conditions, including economic, social, and ecological aspects of sustainable development for knowledge development |
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| 5.2 Economic conditions for knowledge development |
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| 5.3 Identification of needs, structuring and planning of research or development projects |
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| 5.4 Execution of research or development projects |
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| 5.5 Presentation and evaluation of research or development projects |
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