@MISC{NORMSP_3.MIZ,
  AUTHOR = {Nakasho, Kazuhisa and Futa, Yuichi and Shidama, Yasunari},
  TITLE = {Topological Properties of Real Normed Space},
  SECTION1 = {Open and Closed Subsets},
  SECTION2 = {Density},
  SECTION3 = {Separability},
  SECTION4 = {Sequence and Convergence},
  SECTION5 = {Subspaces},
  SECTION6 = {Linear Functions},
  SECTION7 = {Banach Spaces},
  SECTION8 = {Quotient Vector Spaces},
  SECTION9 = {Closure},
  DAY = {15},
  MONTH = {September},
  YEAR = {2014},
  ADDRESS1 = {Shinshu University, Nagano, Japan},
  ADDRESS2 = {Japan Advanced Institute of Science and Technology, Ishikawa, Japan},
  ADDRESS3 = {Shinshu University, Nagano, Japan},
  SUMMARY = {In this article, we formalize topological properties of real
  normed spaces. In the first few parts, open and closed subsets,
  density, separability
  and sequences and their convergence are discussed.
  Then we argue properties of real
  normed subspaces. In the middle of the article, we discuss linear functions
  between real normed spaces. Several kinds of subspaces induced by linear
  functions such as kernel, image and inverse image are dealed with here.
  The fact that Lipschitz continuity operators preserve convergence of sequences
  is also referred here. Then we argue the condition when real normed subspaces
  become Banach spaces. Additionally, we formalize quotient vector space. In the last
  section, we argue the properties of the closure of real normed space.},
  ACKNOWLEDGEMENT = {This work was supported by JSPS KAKENHI 22300285.}}

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

