@MISC{LOPBAN15.MIZ,
  AUTHOR = {Nakasho, Kazuhisa},
  TITLE = {Transformation tools for real linear spaces},
  SECTION1 = {Lipschitz continuity of linear maps from finite-dimensional spaces},
  SECTION2 = {Identification of a real number set with a one-dimensional real normed space},
  SECTION3 = {Identification of real Euclidean space and real normed space},
  SECTION4 = {Transformation to real vector space},
  SECTION5 = {Basis and dimension properties of real linear spaces},
  DAY = {23},
  MONTH = {July},
  YEAR = {2022},
  ADDRESS1 = {Yamaguchi University\\Yamaguchi, Japan},
  MSC2010 = {46A19 46A35 68V20},
  KEYWORDS = {real linear space; real normed space; real Euclidean space; real vector space; basis, dimension},
  SUMMARY = {This paper provides useful tools for working with real linear spaces and real normed spaces. These include the identification of a real number set with a one-dimensional real normed space, the relationships between real linear spaces and real Euclidean spaces, the transformation from a real linear space to a real vector space, and the properties of basis and dimensions of real linear spaces.
    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}
}
