Lectures Spring 2019

Mo 15:00 - 17:00 / 342 HXLY/ & Fri 11:00 - 12:00/ HXLY 341 /

Office Hours (Room: 6M55 )

M345P6&7 : Mo 12:00-13:00 , Tue 13:00-14:00,


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.


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


Kreyszig - Introductory Functional Analysis with Applications


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

CW.1, CW.2,


Homework.2, H2.Solutions

Homework.3, H3.Solutions

Comments, Hints, Solutions

PS.1, PS.2solns, PS.3solns, PS.4solns , Apx_4ps_solns




NOTES 2018

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

Content 2019



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-


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 Lecture Notes



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

James Munkres, Topology (+solutions)

Lebesgue Measure and Integration Theory - T.Tao

Mid-term lecture feedback questions

M345P7 Page by Boguslaw Zegarlinski : Comments, Remarks and Questions Welcome !