@MISC{DUALSP03.MIZ,
  AUTHOR = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
  TITLE = {Weak Convergence and Weak* Convergence},
  ACKNOWLEDGEMENT = {This work was supported by JSPS KAKENHI 22300285 and 23500029.},
  SECTION1 = {Some properties about Dual Spaces of Real Normed Spaces},
  SECTION2 = {Weak Convergence and Weak* Convergence},
  SECTION3 = {Weak Sequential Compactness of Real Banach Spaces},
  DAY = {1},
  MONTH = {July},
  YEAR = {2015},
  ADDRESS1 = {Hirosaki-city, Aomori, Japan},
  ADDRESS2 = {Gifu National College of Technology, Gifu, Japan},
  ADDRESS3 = {Shinshu University\\Nagano},
  SUMMARY = {In this article, we deal with weak convergence on sequences in
  real normed spaces, and weak* convergence on sequences in
  dual spaces of real normed spaces.
  In the first section, we proved some topological properties
  about dual spaces of real normed
  spaces. We used these theorems for the proof of Section 3.
  In Section 2, we defined weak convergence and weak* convergence, and
  proved some properties. And by \verb!RNS_Real! Mizar functor,
  real normed spaces as real number spaces
  already defined in the article \cite{DUALSP02.ABS}, we regarded
  sequences of
  real numbers as sequences of \verb!RNS_Real!. So we proved the last theorem in
  this section using the theorem \cite{LOPBAN_5.ABS:8}.
  In Section 3, we defined weak sequential compactness of real normed spaces.
  We showed some lemmas for the proof and proved the theorem of weak
  sequential compactness of reflexive real Banach spaces.}}

