Special Mathematics Lecture

Contact:
Serge Richard (richard@math.nagoya-u.ac.jp), Rm. 247 in Sci. Bldg. A

Introduction to functional analysis (Spring 2023)

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Registration code : 0053611
Schedule : Wednesday (18.30 - 20.00) in room 207 of Science Building A and on Zoom

• Syllabus : See here

• SML official rule : See here

• NU EMI : Link to the website

• 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
  

• Lecture notes :
  The cumulative notes
  

• 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 n-box, 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^p-norm 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 Cauchy-Schwarz 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 self-adjoint, 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^1-norm, 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, self-adjoint 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 user-friendly introduction to Lebesgue measure and integration