2016 Foundations of Functional Analysis

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Academic unit or major
Computer Science
Uyematsu Tomohiko 
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Course description and aims

The functional analysis is a mathematical tool for signal processing, image processing, pattern recognition, and system theory. The course teaches foundations of functional analysis in an easily-understood manner by using various applications. The aim of the course is to capture various mathematical problems in a unified manner, even though they are seemingly different, by using the concept of space, convergence, and operator.

Student learning outcomes

By completing the course, students will be able to
1) Understand the foundations of functional analysis, and apply to various engineering problems.
2) Understand the projection theorem and fixed-point theorem, and explain their meanings.
3) Interpret the least squares approximation problem geometrically.


linear space, fixed-point theorem for contraction mapping, Banach space, Hilbert space, linear operator, orthogonal projection theorem, least squares approximation problem, reverse problem, generalized Fourier series

Competencies that will be developed

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

Class flow

Every class, the lecturer explains some topics.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Introduction(role of mathematics in engineering) Review linear algebra.
Class 2 Linear space, metric space, set and topology Explain basic mathemtical terms such as the axiom of distance, open set, close set, etc.
Class 3 Completeness, norm, normed space Explain completeness and the axiom of norm.
Class 4 Sequence space Explain the sequence space and its norm by using examples.
Class 5 Functional space Explain the functional space and its norm by using examples.
Class 6 Inner product, inner product space, ‎Schwarz inequality, ‎parallelogram theorem Explain inner product space, ‎Schwarz inequality, and the parallelogram theorem.
Class 7 Banach space Explain typical examples of Banach space.
Class 8 Fixed-point theorem for contract mapping and applications Explain the definition of contraction mapping, and the meaning of the fixed-point theorem.
Class 9 Hilbert space Explain typical examples of Hilbert space.
Class 10 Linear operator Explain the definition of linear operator and its example.
Class 11 Orthogonal decomposition, orthogonal projection theorem Explain the orthogonal decomposition and the orthogonal projection theorem.
Class 12 Orthonormal system Explain the definition of orthonormal system and its properties.
Class 13 Complete orthonormal system Explain the definition of complete orthonormal system and its properties.
Class 14 Generalized Fourier series Explain generalized Fourier series.
Class 15 Linear functional, Riesz's theorem, adjoint operator Explain linear functional, Riesz's theorem and adjoint operator.


(Oクラス)・工学のための関数解析, 山田功 著, 数理工学社, 2009
(Eクラス)・関数解析学の基礎・基本,樋口禎一,芹澤久光,神保敏弥著,牧野 書店, 2001

Reference books, course materials, etc.

Handouts will be distributed at the beginning of class if necessary.

Assessment criteria and methods

Final exam and/or exercises

Related courses

  • Fourier and Laplace Transforms
  • Numerical Analysis
  • Signal Processing
  • Pattern Recognition
  • ICT.A512 : Advanced Information and Communication Theory

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

None required.

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