The fundamentals of the functional analysis are given for the application to the mathematical and computing sciences. Students understand the theory of function spaces such as Lebesgue spaces and Sobolev spaces and the Fourier transform. They are able to apply them to the partial differential equations.
This course emphasizes the importance of rigorous treatment of various problems in mathematical and computing sciences by the use of concepts in the functional analysis. In particular students are able to understand the fundamentals of the operator theory, the Fourier transform and the theory of distributions, and apply them to partial differential equations and so on.
Function spaces, Inequalities for functions, Fourier transform, Distributions, Partial differential equations
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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After preliminaries of topology, the lecture is devoted to the fundamentals to the functional analysis. The Fourier transform and the theory of distributions is given for the applications to the partial differential equations. In order to cultivate a better understanding, some exercises are given.
Course schedule | Required learning | |
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Class 1 | Banach spaces and examples | Understand the contents in the lecture. |
Class 2 | Lebesgue spaces | Understand the contents in the lecture. |
Class 3 | Inequalities in function spaces | Understand the contents in the lecture. |
Class 4 | Convolution and mollifiers | Understand the contents in the lecture. |
Class 5 | Properties of the Fourier transform | Understand the contents in the lecture. |
Class 6 | Fourier inversion formula | Understand the contents in the lecture. |
Class 7 | Properties of rapidly decaying functions | Understand the contents in the lecture. |
Class 8 | Distributions | Understand the contents in the lecture. |
Class 9 | Tempered distributions and the Fourier transform | Understand the contents in the lecture. |
Class 10 | Derivatives of distributions and the Sobolev spaces | Understand the contents in the lecture. |
Class 11 | Sobolev's embedding theorem and Rellich's compactness theorem | Understand the contents in the lecture. |
Class 12 | Convolution of distributions | Understand the contents in the lecture. |
Class 13 | Applications to the Laplace equation | Understand the contents in the lecture. |
Class 14 | Application to the heat equation | Understand the contents in the lecture. |
Class 15 | Applications to the wave equation | Understand the contents in the lecture. |
To be specified in the first lecture.
Unspecified.
Learning achievement is evaluated by reports.
No prerequisites. Students should understand the fundamentals of topology.