Definition and characterization of supersmooth functions on superspace based on Fréchet-Grassmann algebra

Details Definition and characterization of supersmooth functions on superspace based on Fréchet-Grassmann algebra Definition and characterization of supersmooth functions on superspace based on Fréchet-Grassmann algebra

2009-10-20

arxiv.org/abs/0910.3831v2

Physics

Mathematical Physics

32 pages, 2 figures

Scientific paper

Preparing the Fr\'echet-Grassmann (FG-)algebra ${\mathfrak{R}}$ composed with countably infinite Grassmann generators, we introduce the superspace ${\mathfrak{R}}^{m|n}$. After defining Grassmann continuation of smooth functions on ${\mathbb{R}}^m$ to those on ${\mathfrak{R}}^{m|0}$, we introduce a class of functions which are called supersmooth (alias superfields) and are regarded as one of those with countably infinite independent variables. We characterize such supersmooth functions in G\^ateaux (but not necessarily Fr\'echet) differentiable category in Fr\'echet but not in Banach space.

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