@MISC{NEURONS1.MIZ,
  AUTHOR = {Miyajima, Keiichi and Yamazaki, Hiroshi},
  TITLE = {Compactness of Neural Networks},
  SECTION1 = {Preliminaries},
  SECTION2 = {The Ascoli-Arzela Theorem on the Finite 
    Dimensional Normed Linear Spaces},
  SECTION3 = {High order \& Multilayer Perceptron},
  DAY = {30},
  MONTH = {April},
  YEAR = {2022},
  ADDRESS1 = {Ibaraki University \\Ibaraki, Japan},
  ADDRESS2 = {Nagano Prefectural Institute of Technology\\Nagano,     Japan},
  MSC2010 = {46B50 68T05},
  KEYWORDS = {Neural Network, Compactness, Ascoli-Arzela's theorem,
               equicontinuousness of continuous functions,
               equiboundedness of continuous functions },
  SUMMARY = {In this article, Feed-forward Neural Network is formalized.
                 First, the Multilayer perceptron
              \cite{Rosenblatt1958},\cite{Rumelhart1986},\cite{Schmidhuber2015}
              is formalized using function sequence and those synthesis.
                 Next, we show that a set of functions generated by these neural
              networks are equicontinuousness and equiboundedness.
              \cite{YOSHIDA1980},\cite{READSIMON1980}
                 At last, we formalized the compactness of the function set of
              these neural networks by using the Ascoli-Arzela's theorem according to
              \cite{Matuzaka}, \cite{SergeLang}.
             },
  ACKNOWLEDGEMENT = {This work was supported by JSPS KAKENHI Grant Numbers JP17K00182.
                     Prof. Yasunari Shidama, thank you for your cooperation.}}

@BOOK{Rosenblatt1958,
  TITLE={The Perceptron: A Probabilistic Model for Information Storage and Organization
         in the Brain.},
  AUTHOR={Rosenblatt F},
  PUBLISHER={Psychological Review},
  YEAR={1958}
}

@BOOK{Rumelhart1986,
  TITLE = {Learning representations by backpropagating errors.},
  AUTHOR = {Rumelhart DE, Hinton GE, Williams RJ},
  PUBLISHER = {Nature},
  YEAR = {1986}
}

@BOOK{Schmidhuber2015,
  TITLE = {Deep Learning in Neural Networks: An Overview.},
  AUTHOR = {Schmidhuber J}
  PUBLISHER = {Neural Networks},
  YEAR = {2015}
}

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

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

