@MISC{LPSPACE2.MIZ,
  AUTHOR = {Watase, Yasushige and Endou, Noboru and Shidama, Yasunari},
  TITLE = {On $L^p$ Space Formed by Real-valued Partial Functions},
  SECTION1 = {Preliminaries of Powers of Numbers and Operations of
    Real Sequences},
  SECTION2 = {Real Linear Space of $L^p$ Integrable Functions},
  SECTION3 = {Preliminaries of Real Normed Space of $L^p$
    Integrable Functions},
  SECTION4 = {Real Normed Space of $L^p$ Integrable Functions},
  SECTION5 = {Preliminaries of Completeness of $L^p$ Space},
  SECTION6 = {Relations between $L^1$ Space and $L^p$ Space},
  DAY = {4},
  MONTH = {February},
  YEAR = {2010},
  ADDRESS1 = {Graduate School of Science and Technology\\Shinshu University\\4-17-1 Wakasato, Nagano, Japan},
  ADDRESS2 = {Gifu National College of Technology\\Gifu, Japan},
  ADDRESS3 = {Faculty of Engineering\\Shinshu University\\4-17-1 Wakasato, Nagano, Japan},
  SUMMARY = {This article is the continuation of \cite{LPSPACE1.ABS}.
  We define the set of $L^p$ integrable functions -- the set of all partial
  functions whose absolute value raised to the $p$-th power is integrable.
  We show that $L^p$ integrable functions form the $L^p$ space.
  We also prove Minkowski's inequality, H\"{o}lder's inequality and that
  $L^p$ space is Banach space (\cite{Halmos:1974}, \cite{Rudin:1987}).}}

@BOOK{Halmos:1974,
 AUTHOR={Paul R. Halmos},
 TITLE={Measure Theory},
 PUBLISHER={Springer-Verlag},
 YEAR=1974}

@BOOK{Rudin:1987,
 AUTHOR={Rudin, Walter},
 TITLE= {Real and complex analysis},
 PUBLISHER={McGraw-Hill, Inc.}
 YEAR={1987}
