Definition and characterization of supersmooth functions on superspace based on Fr\'echet-Grassmann algebra

Preparing the Fr\'echet-Grassmann (FG-)algebra ${\fR}$ composed with countably infinite Grassmann generators, we introduce the superspace ${\fR}^{m|n}$. After defining Grassmann continuation of smooth functions on ${\euc}^m$ to those on ${\fR}^{m|0}$, we introduce a class of functions on ${\fR}^{m|n}$ which are called supersmooth. In this paper, we characterize such supersmooth functions in G\^ateaux (but not necessarily Fr\'echet) differentiable category on Fr\'echet but not on Banach space. This type of arguments for $G^{\infty}$-functions is mainly done on the Banach-Grassmann (BG-)algebra, but we find it rather natural to work within FG-algebra when we treat systems of PDE such as Dirac, Weyl or Pauli equations. In that application, we need to prove that the solution of the (super) Hamilton equation is supersmooth w.r.t. initial data. Though we took this point of view in our previous works, but is managed rather insufficiently. Therefore, we re-treat this subject here to answer affirmatively. We give also local or global inverse function theorems for supersmooth functions on ${\fR}^{m|n}$.