@MISC{ASCOLI2.MIZ,
  AUTHOR = {Miyajima, Keiichi and Yamazaki, Hiroshi},
  TITLE = {Ascoli-{A}rzela's Theorem (Metric Space Version)},
  SECTION1 = {Equicontinuousness and Equiboundedness of Continuous Functions},
  SECTION2 = {Ascoli-Arzela's Theorem},
  DAY = {28},
  MONTH = {November},
  YEAR = {2024},
  ADDRESS1 = {Ibaraki University \\Ibaraki, Japan},
  ADDRESS2 = {Nagano Prefectural Institute of Technology\\Nagano, Japan},
  MSC2010 = {46B50},
  KEYWORDS = {Ascoli-Arzela's theorem, equicontinuousness of continuous functions, 
               equiboundedness of continuous functions},
  SUMMARY = {In this article, the Ascoli-Arzela's theorem on metric space is formalized \cite{SergeLang},
                  \cite{Matuzaka}, \cite{ReadSimon1980}.
              First, we gave definitions of equicontinuousness and equiboundedness 
              of a set of continuous functions
                  \cite{YOSHIDA1980}, \cite{Ozawa}, \cite{driver2003},
                  \cite{Schwartz1997a}, \cite{Schwartz1997b}.
              Next, we formalized the Ascoli-Arzela's theorem using those definitions, 
              and proved this theorem. 
              From this result, Ascoli-Arzela's theorem can be applied in a metric space that is easier to apply.
             },
  ACKNOWLEDGEMENT = {The authors would like to express our gratitude to Prof. 
   Yasunari Shidama for his support and encouragement.}}

@BOOK{Schwartz1997a, 
  TITLE={Th\'{e}orie des ensembles et topologie, tome 1. Analyse},
  AUTHOR={Laurent Schwartz},
  PUBLISHER={Hermann},
  YEAR={1997},
  MONTH={10},
}

@BOOK{Schwartz1997b, 
  TITLE = {Calcul diff\'{e}rentiel, tome 2. Analyse},
  AUTHOR = {Laurent Schwartz},
  PUBLISHER = {Hermann},
  YEAR = {1997},
  MONTH = {10}
}

@BOOK{driver2003,
  TITLE = {Analysis Tools with Applications},
  AUTHOR = {Bruce K. Driver}
  PUBLISHER = {Springer, Berlin},
  YEAR = {2003},
  MONTH = {10}
}

@BOOK{YOSHIDA1980,
      AUTHOR = {Yoshida, Kosaku},
      TITLE = {Functional Analysis},
      PUBLISHER = {Springer-Verlag},
      YEAR = {1980}}

@BOOK{ReadSimon1980,
      AUTHOR = {Read, Michael and Simon, Barry},
      TITLE = {Functional Analysis (Methods of Modern Mathematical Physics)},
      PUBLISHER = {Academic Press},
      YEAR = {1980}}

@BOOK{SergeLang,
      AUTHOR = {Lang, Serge},
      TITLE = {Real and Functional Analysis (Texts in Mathematics)},
      PUBLISHER = {Springer-Verlag},
      YEAR = {1993}}

@BOOK{Matuzaka,
      AUTHOR = {Matsuzaka, Kazuo},
      TITLE = {Sets and Topology (Introduction to Mathematics)},
      PUBLISHER = {IwanamiShoten},
      YEAR = {2000}}

@article{Ozawa,  
  TITLE = {{Ascoli-Arzel\`{a} theorem}},
  AUTHOR = {Ozawa, Tohru},
  URL = {http://www.ozawa.phys.waseda.ac.jp/pdf/Ascoli.pdf},
  YEAR = {2012},
  PDF = {http://www.ozawa.phys.waseda.ac.jp/pdf/Ascoli.pdf}}

