M345P7 FUNCTIONAL ANALYSIS

Lectures Autumn 2019

Tue 16:00 - 18:00 / 341 HXLY/ / HXLY 130 (Wk 11) & Thu 10:00 - 11:00/ HXLY 308

Office Hours (Room: 6M55 )

M345P7 : Tue 12:00-13:30,


Syllabus

This course brings together ideas of continuity and linear algebra. It concerns vector spaces with a distance, and involves linear maps; the vector spaces are often spaces of functions.

Vector spaces. Existence of a Hamel basis. Normed vector spaces. Banach spaces. Finite dimensional spaces. Isomorphism. Separability. The Hilbert space. The Riesz-Fisher Theorem. The Hahn-Banach Theorem. Principle of Uniform Boundedness. Dual spaces. Operators, compact operators. Hermitian operators and the Spectral Theorem. Extension of the above topics. Banach algebras.

Short Content 2019

Basic Bibliography

Imperial College Library Electronic Resources :

Rabindranath Sen, A First Course in Functional Analysis Theory and Applications 
Brian P. Rynne and Martin A. Youngson, Linear Functional Analysis 
Barbara MacCluer, Elementary Functional Analysis 
Haïm Brezis, Functional analysis, Sobolev spaces, and partial differential equations 
Marián Fabian,Petr Habala,Petr Hájek, Vicente Montesinos,Václav Zizler, 
Banach Space Theory, The Basis for Linear and Nonlinear Analysis

Theorie des Operations Lineares by Stefan Banach

@archive.org

Kreyszig - Introductory Functional Analysis with Applications


Current Reading

I.

A First Course in Functional Analysis:Theory and Applications, Rabindranath Sen Chapter.1

Erwin Kreyszig - Introductory Functional Analysis with Applications Chapter.1

II.

Sergei Ovchinnikov, Functional Analysis An Introductory Course ; Chapter 3: Special Spaces
 
https://link.springer.com/book/10.1007%2F978-3-319-91512-8

Erwin Kreyszig - Introductory Functional Analysis with Applications Chapter.2 : 2.1 - 2.3

Marián Fabian,Petr Habala,Petr Hájek, Vicente Montesinos,Václav Zizler, 
Banach Space Theory, The Basis for Linear and Nonlinear Analysis 
Examples of Schauder Basis pp. 185-187
III. 

Erwin Kreyszig - Introductory Functional Analysis with Applications Chapter.3

Brian P. Rynne and Martin A. Youngson, Linear Functional Analysis Chapter.3

Inner Product Spaces, Hilbert Spaces Pages 51-85

IV. 

Erwin Kreyszig - Introductory Functional Analysis with Applications Chapter.2.4-2.10

Marián Fabian,Petr Habala,Petr Hájek, Vicente Montesinos,Václav Zizler, 
Banach Space Theory, The Basis for Linear and Nonlinear Analysis 
Section 1.3                  
V. 

Erwin Kreyszig - Introductory Functional Analysis with Applications Chapter.4.1-4.5

Brian P. Rynne and Martin A. Youngson, Linear Functional Analysis Chapter.5.1-5.3
Marián Fabian,Petr Habala,Petr Hájek, Vicente Montesinos,Václav Zizler, 
Banach Space Theory, The Basis for Linear and Nonlinear Analysis 
Section 2.1-2.2     
VI. 

Haïm Brezis, Functional analysis, Sobolev spaces, and partial differential equations Chapter.2

Rabindranath Sen, A First Course in Functional Analysis Theory and Applications Chapter.VII
Barbara MacCluer, Elementary Functional Analysis Chapter.3

VII.      
Rabindranath Sen, A First Course in Functional Analysis Theory and Applications Chapter.8

Erwin Kreyszig - Introductory Functional Analysis with Applications Chapter.8.3-8.4                              

Current Content

FANA - Chapter.1

PROBLEM SETS

PS.1, PS.2 , PS.3 , PS.4, PS.5

CW.1, CW.2, CW.3

Comments, Hints, Solutions

PS.1, PS.2solns , PS.3solns , PS.4solns , PS.5solns

 

Mastery Material FANA2019

TASKS

T_w.1

T_w.2

T_w.3

T_w.4

T_w.5

NOTES 2019

notes.1, notes.2, notes.3 , notes.4

NOTES 2018

notes.1, notes.2, notes.3


SOME LINKS

*Home page for Axiom of Choice

*Abstract linear spaces

*Banach Spaces

*Hahn-Banach Theorem

*Banach-Steinhaus Theorem

*Banach Fixed Point Theorem

*Banach - Tarski Paradox

*The Banach Space Bulletin Board

*Virtual Science Library: Studia Mathematica 1929-

encyclopediaofmath.org

Banach, Stefan; Steinhaus, Hugo (1927), "Sur le principe de la condensation de singularités" (PDF),
Fundamenta Mathematicae,
9: 50–61. (in French)

Banach-Steinhaus by gliding hump method

Terry J. Morrison, Functional Analysis: An Introduction to Banach Space Theory, 
2001 John Wiley & Sons, p.77. 
Albrecht Pietsch, History of Banach Spaces and Linear Operators,2007 Birkhauser Boston, 
p.41 
Sokal,Alan (2011),"A really simple elementary proof of the uniform
boundedness theorem", Amer. Math. Monthly, 118: 450–452, arXiv:1005.1585,
doi:10.4169/amer.math.monthly.118.05.450.

Banach, Stefan; Tarski, Alfred (1924). "Sur la décomposition des ensembles de points en parties respectivement congruentes" .
Fundamenta Mathematicae (in French). 6: 244–277.

Pawlikowski, Janusz (1991). "The Hahn-Banach theorem implies the Banach-Tarski paradox". Fundamenta Mathematicae. 138: 21–22.

Hilbert XIII@wiki & MathEncyclopedia

Per Enflo : Not every separable Banach space has a Schauder basis.

M.G. Nadkarni and V.S. Sunder Hamel bases and measurability

History of Mathematics

*Stefan Banach

Stefan Banach by Hugo Steinhaus

Banach Polish Mathematic Genius_@uTube

*Scottish Book

* Juliusz Pawel Schauder

*Hugo Steinhaus

Albrecht Pietsch , History of Banach Spaces and Linear Operators

Frigyes Riesz, Andrey Kolmogorov, Alfred Tarski

Some External Lecture Notes

mit_open_courseware

eth~salamon

Hamel Basis

Open Library : Banach Spaces

Classical Banach Spaces I: Sequence Spaces by Joram Lindenstrauss, Lior Tzafriri

Valery Serov: Fourier Series, Fourier Transform and Their Applications to Mathematical Physics


Lebesgue Measure and Integration Theory – T.Tao

James Munkres, Topology (+solutions)









Mid-term lecture feedback questions


 

M345P7 Page by Boguslaw Zegarlinski

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