@MISC{DUALSP01.MIZ,
  AUTHOR = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
  TITLE = {Dual Spaces and Hahn-Banach's Theorem},
  ACKNOWLEDGEMENT = {This work was supported by JSPS KAKENHI 22300285,
  23500029.},
  SECTION1 = {Dual Spaces of Real Linear Spaces},
  SECTION2 = {Dual Spaces of Real Normed Spaces},
   SECTION3 = {Hahn-Banach's Extension Theorem},
  DAY = {31},
  MONTH = {March},
  YEAR = {2014},
  ADDRESS1 = {Hirosaki-city, Aomori, Japan},
  ADDRESS2 = {Gifu National College of Technology, Gifu, Japan},
  ADDRESS3 = {Shinshu University\\Nagano},
  SUMMARY = {In this article, we deal with
  dual spaces and Hahn-Banach's Theorem.
  At the first, we defined dual spaces of real linear spaces
  and proved related basic properties.
  Next, we defined dual spaces of real normed spaces.
  We formed the definitions based on dual spaces of real linear spaces.
  In addition, we proved properties of the norm about elements of dual
  spaces. For the proof we referred to descriptions in the article
  \cite{LOPBAN_1.ABS}.
  At the last, applying theorems of the second section,
  we proved Hahn-Banach's extension theorem in real normed
  spaces. We used extensively \cite{HAHNBAN.ABS}.}}


