Course title
6M0037001
Functional Analysis

 idogawa tomoyuki Click to show questionnaire result at 2018
Course content
In this lecture, students will study the theorems of functional analysis deeply and learn how to apply them to numerical analysis.
The studentes are expected to know the basic concepts concerning topological space, Banach space, Hilbert space, Lebesgue integral, etc. So the lecture will be concentrated on studying how to apply the fundamental theories of functional analysis to the interpolation theorems or numerical analysis of PDE.
Purpose of class
To understand how to apply the fundamental theories of functional analysis to the interpolation theorems or numerical analysis.
Goals and objectives
  1. To learn the relation between functional and numerical analysis, and understand how to jusify approximate solutions which are calcurated numerically.
  2. To learn the uniform bounded principle and its applications, and understand the theorems of convergence from the viewpoint of functional analysis.
  3. To learn applications of the uniform bounded principle to PDE, and understand the error estimate or stability theories of numerical solution.
Language
Japanese
Class schedule

 Class schedule HW assignments (Including preparation and review of the class.) Amount of Time Required
1． Functional analysis and numerical analysis (introduction). Review of the undergraduate mathematics (derivatives/integrals, numerical analysis, and functional analysis). 190minutes
2． Newton method for infinite-dimensional spaces (preliminaries; existence lemma of inverse operator). Review of the undergraduate mathematics (matrices and infinite series). 190minutes
3． Newton method for infinite-dimensional spaces (Kantrovich's theorem) . Review of the undergraduate mathematics (Newton's method on finite dimensional spaces). 190minutes
4． Review on topological and normed spaces. Review of the undergraduate mathematics (sets and topology). 190minutes
5． Numerical analysis on Banach spaces of finite dimension. Review of the undergraduate mathematics (linear algebra). 190minutes
6． Numerical analysis on Banach spaces of infinite dimension. Review of the undergraduate mathematics (functional analysis). 190minutes
7． Operator norms. Review of the undergraduate mathematics (functional analysis). 190minutes
8． Uniform bounded principle. Review of the previous lecture. 190minutes
9． Theory of row transformation. Review of the previous lecture. 190minutes
10． Application of uniform bounded principle to row transformations. Review of the previous lecture. 190minutes
11． Application to interpolation theories (polynomial interpolation). Review of the previous lecture. 190minutes
12． Application to interpolation theories (spline interpolation). Review of the previous lecture. 190minutes
13． Application to PDE. Review of the previous lecture. 190minutes
14． Summary/Final examination. Review of the all of lectures. 190minutes
Total. - - 2660minutes
Relationship between 'Goals and Objectives' and 'Course Outcomes'

 Examinations Total.
1. 40% 40%
2. 40% 40%
3. 20% 20%
Total. 100% -
Evaluation method and criteria
Based on the result of final examination.
Textbooks and reference materials
Reference: 数値関数解析の基礎, 鈴木千里, 森北出版, 2001. (in Japanese)
Prerequisites
Students are expected to understand very well about calculus, linear algebra and numerical analysis studied in undergraduate course. Moreover, it is required that the subjects Set theory and topology, Functional analysis I/II (or similar ones) have been already taken.
Office hours and How to contact professors for questions
  • Tuesday 12:15--13:00.
Relation to the environment
Non-environment-related course
Regionally-oriented
Non-regionally-oriented course
Development of social and professional independence
  • Non-social and professional independence development course
Active-learning course
N/A
Course by professor with work experience
Work experience Work experience and relevance to the course content if applicatable
N/A N/A
Last modified : Thu Mar 21 15:28:43 JST 2019