### 2016　Functional Analysis

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Instructor(s)
Yanagida Eiji
Course component(s)
Lecture
Day/Period(Room No.)
Mon7-8(H116)  Wed3-4(H116)
Group
-
Course number
MTH.C351
Credits
2
2016
Offered quarter
3Q
Syllabus updated
2017/1/11
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

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.

### Student learning outcomes

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 that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

Standard lecture course

### Course schedule/Required learning

Course schedule Required learning
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.
Class 15 applications to differential equations, comprehension check-up Details will be provided during each class session.

### Textbook(s)

"Functional Analysis", Kyuya Masuda, Shokabo

### Reference books, course materials, etc.

None in particular

### Assessment criteria and methods

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 (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed Real Analysis I, Real Analysis II and Exercises in Analysis C I.

### Other

None in particular