@MISC{LOPBAN14.MIZ,
  AUTHOR = {Nakasho, Kazuhisa and Futa, Yuichi},
  TITLE = {Isomorphism between Spaces of Multilinear Maps and   
  Nested Compositions over Real Normed Vector Spaces},
  SECTION1 = {Preliminaries},
  SECTION2 = {Spaces of multilinear maps and nested 
   compositions   over real normed vector spaces}
  DAY = {30},
  MONTH = {April},
  YEAR = {2022},
  ADDRESS1 = {Yamaguchi University\\Yamaguchi, Japan},
  ADDRESS2 = {Tokyo University of Technology\\Tokyo, Japan},
  MSC2010 = {15A69 47A07 68V20},
  KEYWORDS = {Banach space; composition function; multilinear function},
  SUMMARY = {This paper formalizes that the isometric isomorphisms between 
  spaces formed by an (n+1)-dimensional multilinear map and an 
  n-fold composition of linear maps on real normed spaces. 
  This result is used to describe the space of nth-order derivatives of 
  the Frechet derivative as a multilinear space. In Chapter 1, we discuss 
  the spaces of 1-dimensional multilinear maps and 0-fold 
  compositions as a preparation, and in Chapter 2, we extend the 
  discussion to the spaces of (n+1)-dimensional multilinear 
  map and an n-fold compositions.
  We referred to \cite{miyadera:1972}, \cite{yoshida:1980}, 
  \cite{Schwartz1997a}, \cite{Schwartz1997b} in this formalization.}
  ACKNOWLEDGEMENT = {I would like to express my gratitude to Professor Emeritus 
  Yasunari Shidama for his helpful advice. This work was supported by 
  JSPS KAKENHI Grant Number JP20K19863.}}

@BOOK{miyadera:1972,
 AUTHOR={Isao Miyadera},
 TITLE={Functional Analysis},
 PUBLISHER={Riko-Gaku-Sya},
 YEAR=1972}

@BOOK{yoshida:1980,
 AUTHOR={Kosaku Yoshida},
 TITLE={Functional Analysis},
 PUBLISHER={Springer},
 YEAR=1980}

@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}
}
