Special Mathematics Lecture
Contact:
Serge Richard (richard@math.nagoyau.ac.jp), Rm. 247 in Sci. Bldg. A
Introduction to functional analysis (Spring 2023)
Registration code : 0053611
Schedule : Wednesday (18.30  20.00) in room 207 of Science Building A and on Zoom

SML official rule :
See here

Class dates :
April 12, 19, 26
May 2, 10, 17, 24, 31
June 7, 14, 21, 28
July 5, 12

Program :
1) Distribution theory
2) Lebesgue theory of integration
3) Operator theory on Hilbert spaces

Weekly summaries :
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14

Study sessions :
Will be organized on an individual basis by some international students

Evaluation :
You need to submit the solutions of some exercises and/or the proofs of some statements.
These submissions can take place at any time during the semester.
If you have any question, contact me
or the TA Diep Minh Nguyen

Student's reports :
On the exchange of partial derivatives, by Haruki Tsunekawa
Distributions: characterisation, support, and order, by Pratham Dhomne and Vic Austen
About some distributions, by Yat Ming Luk
Supports of regular and dirac delta distributions, by Yat Ming Luk
Derivatives of regular distributions, by Alberto Thornton
Proofs on some distributions, by Firdaus Rafi Rizqy, Hadiko Rifqi Aufa Sholih, and Sekiya Emika
On regular distributions, by Haruka Yajima
Convergence of derivatives of distributions, by Yuu Hiramatsu
Three standard distributions, by Yuu Hiramatsu
On the convergence of distributions + an improper Riemann integral obtained with Cauchy's integral theorem, by Vic Austen
Properties of Fourier transform, by Yat Ming Luk
On various distributions, by Zhang Jiabin
Standard distributions, by Ryosuke Mizutani
Sequence of functions converging to the 0 function
whose derivatives do not converge to 0, by Zhang Zhiyang
Derivative of a regular distribution, and order of some distributions, by Ryosuke Mizutani
On the functions 1/x and ln(x), by Koichi Kato
About the distribution Pv 1/x, by Koichi Kato
Convergence of distribution, including sin(jx)/x, by Yamada Ai
Schwartz functions are integrable, by Chenxi Zeng
Schwartz space and the Fourier transform of some important tempered distributions, by Tue Tai Nguyen
On the derivative of the Dirac delta distribution, by Katayama Marin
Reminder on topology and test functions, by Tarumizu Rintaro
The distribution Pv 1/x, by Haruka Yajima
Properties of Fourier transform, by Anna Stollenwerk
Outer measure: volume of a nbox, by Yat Ming Luk
On outer and inner Lebesgue measures, by Vic Austen
The Lebesgue measure of rational numbers is 0, by Yamamura Yuta
Properties of Fourier transform, with an emphasis on F D = X F, by Firdaus Rafi Rizqy and Sekiya Emika
Example of a non Riemann integrable function, by Katayama Marin and Alberto Thornton
Derivative of x in the sense of distributions, by Katayama Marin and Alberto Thornton
Equivalent definitions of Lebesgue measurability of a function, by Firdaus Rafi Rizqy and Sekiya Emika
Equivalence relation defined by equality almost everywhere, by Yat Ming Luk
Dirichlet function, by Yat Ming Luk
Application of the dominated convergence theorem for the convergence to the Dirac delta distribution, by Ryosuke Mizutani
Equivalence class of functions equal almost everywhere, by Yamada Ai
The Dirichlet function is not Riemann integrable, by Masumi Okamoto
Minkowski inequality, by Yat Ming Luk
About properties of Fourier transform and measurability of functions, by Zhang Jiabin
Limsup, liminf, and limit, by Yamamura Yuta
About Lebesgue measurablity of characteristic functions, by Ryosuke Mizutani
Reminder on Lebesgue measurable sets, by Tarumizu Rintaro
Norm on L^1, by Sekiya Emika
Any orthocomplement is a closed subspace, by Yamada Ai
Proof that test functions are dense in L^2 by using a convolution technique, by Zhang Jiabin
Motivation for introducing distributions, by Tetta Watari
About outer Lebesgue measure, by Tetta Watari
A Proof of Hölder inequality for counting measure, by Zhou Yifan
Subadditivity of outer Lebesgue measure, by Ryosuke Mizutani
The convergence ol L^pnorm when p goes to infinity, by Zhou Yifan
About splines: functions locally defined by polynomials, by Dominik Strutz
The orthocomplement of any subset is a closed subspace, by Firdaus Rafi Rizqy and Sekiya Emika
Proofs of Hölder and Minkowski inequalities, by Hadiko Rifqi Aufa Sholih
Lebesgue measurability, Lebesgue integrability, limsup and liminf functions, by Tetta Watari
Proof of Riesz lemma, by Hadiko Rifqi Aufa Sholih
Five properties of Lebesgue measure, by Ryosuke Mizutani
Proofs of Hölder and Minkowski inequalities, by Haruki Tsunekawa
Norm on a Hilbert space, by Alberto Thornton
Monotone functions are Riemann integrable, by Yoshida Koki
Proof of CauchySchwarz inequality, by Sirawich Saranakomkoop
Proof of inequalities on Hilbert space, by Yamamura Yuta
For bounded functions, Lesbesgue integrability and Lebesgue measurability are equivalent, by Tetta Watari
L^p spaces with p smaller than 1, by Adam Matefy
Pointwise convergence without L^1 convergence, by Ryosuke Mizutani
Proofs of some useful inequalities valid in Hilbert spaces, by Zachary Kokot
On inequalities in Hilbert spaces, by Anna Stollenwerk
The interior and closure for a set of matrices of a certain rank, by Zhou Yifan
About Riesz lemma and Neumann series, by Haruki Tsunekawa
Proofs of some relations for orthogonal projections, by Zachary Kokot
Proofs of inequalities in Hilbert spaces, by Yoshida Koki
Orthogonal systems in Hilbert space and applications, by Tue Tai Nguyen
Properties of Lebesgue measurable sets, by Anna Stollenwerk
About Lebesgue outer measure, by Dominik Strutz
On multiplication operators: adjoint and selfadjoint, by Firdaus Rafi Rizqy and Sekiya Emika
Sequence weakly converging to 0 but not strongly convergent, by Zhang Zhiyang
On the orthocomplement of a subspace, by Zhang Zhiyang
Four inequalities in a Hilbert space, by Masumi Okamoto
The Lebesgue outer measure of a closed box, by Masumi Okamoto
Prove a function to be Riemann integrable, by Zhou Yifan
The diameter of a bounded set is equal to its boundary in a normed space, by Zhou Yifan
The interior and closure of a set change with the norm, by Zhou Yifan
The L^1norm, by APRU students and TAs
About compact operators, by Chenxi Zeng
Eigenvalues of multiplication operators, by Adam Matefy
Orthogonal projections, by Adam Matefy
About convergences on Hilbert spaces and on bounded operators, by Dominik Strutz
Bijective relations between spectral measures, spectral families,
selfadjoint operators, and strongly continuous unitary groups, by Firdaus Rafi Rizqy
Jensen's inequality, and an application, by Yuu Hiramatsu

References: (available upon request)
[Am] W. Amrein, Hilbert space methods in quantum mechanics
[Di] G. van Dijk, Distribution theory: convolution, Fourier transform, and Laplace transform
[Ne] F.S. Nelson, A userfriendly introduction to Lebesgue measure and integration
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