@MISC{NDIFF_5.MIZ,
  AUTHOR = {Shidama, Yasunari},
  TITLE = {Differentiable Functions on Normed Linear Spaces},
  SECTION1 = {Preliminaries},
  SECTION2 = {Mean Value Theorem for Vector-valued Functions},
  SECTION3 = {Partial Derivative of a Function of Several Variables},
  SECTION4 = {Linearity of Partial Differential Operator},
  SECTION5 = {Continuous Differentiatibility of Partial Derivative},
  DAY = {2},
  MONTH = {June},
  YEAR = {2011},
  ADDRESS1 = {Shinshu University, Nagano, JAPAN},
  SUMMARY = {In this article, we formalize differentiability of
  functions on normed linear spaces.
  Partial derivative, mean value theorem for vector-valued functions,
  continuous differentiability, etc. are formalized.
  As well known, there is no exact analog of the mean
  value theorem for vector-valued functions.
  However a certain type of generalization of the mean
  value theorem to for vector-valued functions is obtained as follows:
  If $||.D?(x + th).||$ is bounded for $t$ between 0 and 1
  by some constant $M,$ then  $||. f(x + th) - f(x).|| <= M ||.h.||.$
  This theorem is called the mean value theorem for vector-valued functions.
  By this theorem, the relation between the (total) derivative and
  the partial derivatives of a function is derived \cite{Schwartz}.},
  ACKNOWLEDGMENT = {This work was supported by JSPS KAKENHI 22300285.}}

@ARTICLE{Schwartz,
      AUTHOR  = {Schwartz, Laurent},
      TITLE   = {Cours d'Analyse},
      JOURNAL = {Hermann, Paris},
      YEAR    = 1967}
