2025 (Current Year) Faculty Courses School of Science Undergraduate major in Mathematics
Functional Analysis
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Michiaki Onodera
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.C351
- Number of credits
- 200
- Course offered
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The main subject of this course is about basic concepts in infinite dimensional linear spaces and linear operators between them. After introducing basic concepts of infinite dimensional linear spaces and linear operators (normed spaces, Banach spaces, Hilbert spaces, bounded operators), we explain differences between finite and infinite dimensional spaces and learn their fundamental properties. Next we introduce concept of duality and learn its role in infinite dimension through representation theorem of linear functional and weak topology. Finally, we explain spectral theory of compact self-adjoint operators and learn their applications through concrete problems.
Functional analysis studies the algebraic, geometric and analytic structures of infinite dimensional spaces and operators acting on these spaces. This course covers the basic facts of linear functional analysis and their applications. Students will have the chance to see practical problems are solved elegantly by applying abstract notion and theorems from functional analysis.
Course description and aims
Students are expected to understand the following
・Importance of linear and topological structures in infinite dimensional spaces.
・Basic properties of Banach spaces and bounded linear operators.
・Geometric structure of Hilbert spaces.
・Importance of Banach's three big theorems.
・Concept of duality and its significant role in infinite dimension.
・Importance of compactness in infinite dimension through spectral theory of compact operators.
・Practical problems are solved elegantly by applying abstract notion and theorems
Keywords
normed spaces, Banach spaces, Hilbert spaces, linear operators, Banach's theorems. dual spaces, resolvent, spectrum, compact operators
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | vector spaces, normed spaces | Details will be provided during each class session. |
| Class 2 | Banach spaces, completion, examples | Details will be provided during each class session. |
| Class 3 | linear operators, bounded linear operators, closed operators, examples | Details will be provided during each class session. |
| Class 4 | inverse operators, Neumann series, examples | Details will be provided during each class session. |
| Class 5 | inner product spaces, Hilbert spaces, orthogonal projections, projection theorem | Details will be provided during each class session. |
| Class 6 | Fourier series, Bessel's inequality, complete orthonormal system, Parseval's relation | Details will be provided during each class session. |
| Class 7 | Open mapping theorem, closed graph theorem | Details will be provided during each class session. |
| Class 8 | uniform-boundedness principle, examples | Details will be provided during each class session. |
| Class 9 | dual spaces, conjugate spaces, weak topology, weak convergence | Details will be provided during each class session. |
| Class 10 | conjugate operators, self-adjointness, integral kernel of Hilbert-Schmidt type | Details will be provided during each class session. |
| Class 11 | Hahn-Banach theorem, topological complementary subspace | Details will be provided during each class session. |
| Class 12 | compact operators, Ascoli-Arzela's theorem, examples | Details will be provided during each class session. |
| Class 13 | spectrum, resolvent | Details will be provided during each class session. |
| Class 14 | Riesz theory, alternative theorem | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
Not required
Reference books, course materials, etc.
"Functional Analysis", Shigetoshi Kuroda, Kyoritsu Shuppan
"Functional Analysis", Kyuya Masuda, Shokabo
Evaluation methods and criteria
Based on overall evaluation of the results for report and final examinations. Details will be announced during a lecture.
Related courses
- MTH.C305 ： Real Analysis I
- MTH.C306 ： Real Analysis II
- MTH.C211 ： Applied Analysis I
- MTH.C212 ： Applied Analysis II
- ZUA.C306 ： Exercises in Analysis C I
Prerequisites
Students are expected to have passed Real Analysis I, Real Analysis II and Exercises in Analysis C I.
Other
None in particular