In this lecture, I will explain basic facts of functional analysis. Functional analysis is a theory for functional spaces
and linear operators, and it is often called as linear algebra in infinite-dimensional spaces. It is applied to various sciences,
for example, differential equations, numerical analysis, stochastic analysis, and quantum mechanics. Functional equations,
e.g. differential equations and integral equations, are extensively studied from the viewpoint of functional analysis because
these equations are regarded as equations of operators on the functional spaces. In addition, I strongly recommend that students
who are interested in functional analysis should take "Analysis III" after this class.
In general, it is hard to find an exact solution for a functional equation. But, we can not say that the equation has no solution:
We may have no elementary functions for expressing the solution; or the equation has really no solution. Functional Analysis
is a theory to study the existence/non-existence of solutions of functional equations. When you try to find a solution of
functional equation with no solution by a computer, you unfortunately get "a function", not the solution. Students of mathematical
sciences and of the teacher-training course should be aware of such a fake.
- You can describe the definition and examples of Banach space.
- You can describe the definitions and examples of Hilbert space and complete orthonormal system.
- You can describe the character of linear functional and linear operator.
| Class schedule | HW assignments (Including preparation and review of the class.) | Amount of Time Required | |
|---|---|---|---|
| 1. | Linear spaces | Check out "linear spaces" in advance. | 190minutes |
| 2. | Principle of contraction mapping | Check out "metric spaces" and "completeness" in advance. | 190minutes |
| 3. | Normed spaces, Banach spaces | Check out "completeness" in advance. | 190minutes |
| 4. | Example of Banach spaces: R^n, C^n, l^p | Check out "Euclid spaces" and "infinite series" in advance. | 190minutes |
| 5. | Example of Banach spaces: C[a,b] | Check out "continuity" in advance. | 190minutes |
| 6. | Attention on Mid-term assignment | Check out Sections 1-6 in advance. | 190minutes |
| 7. | Inner product spaces, Hilbert spaces | Check out "inner products" in advance. | 190minutes |
| 8. | Orthogonal decomposition, Projection operator | Check out "direct sum decomposition" in advance. | 190minutes |
| 9. | Complete orthonormal systems | Check out "basis" in advance. | 190minutes |
| 10. | Linear functional, Conjugate spaces, Riesz's theorem | Check out "linearity", "linear functional" and "inner product" in advance. | 190minutes |
| 11. | Linear operators: boundedness and continuity | Check out "linear mappings" in advance. | 190minutes |
| 12. | Lebesgue integral (1) measure theory | Check out "linear operators" in advance. | 190minutes |
| 13. | Lebesgue integral (2) integral theory, L^p | Check out "bounded linear operators" in advance. | 190minutes |
| 14. | Attention on Final assignment | Check out all sessions in advance. | 190minutes |
| Total. | - | - | 2660minutes |
| Mid-term assignment | Final assignment | Total. | |
|---|---|---|---|
| 1. | 50% | 50% | |
| 2. | 25% | 25% | |
| 3. | 25% | 25% | |
| Total. | 50% | 50% | - |
Mid-term assignment and Final assignment. As a criterion, if you determine that you understand 60% of the content covered
in the class, your final score will be 60 points.
Higuchi, Serizawa and Jimbo, "KANSUKAISEKIGAKU NO KISO/KIHON", Makino Shoten.
- Course that cultivates an ability for utilizing knowledge
| Work experience | Work experience and relevance to the course content if applicable |
|---|---|
| N/A | N/A |
Last modified : Thu Apr 01 04:03:58 JST 2021