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Extra info for A nondegeneracy result for a nonlinear elliptic equation
Example text
It is the lack of invertibility of the anticommuting generators which makes such a topology useful, because it has the effect that terms containing a given generator in a factor cannot contribute to terms without that generator in the product. The corresponding situation with a Clifford algebra would be quite different. 2 The topology of superspace There are a number of different topologies one can use on the superspace Rm,n S . The most important topology is that introduced by DeWitt [43]; despite the fact that it is a non-Hausdorff topology, it will emerge in the following chapter that specific algebraic features make it the appropriate topology to use in many aspects of supermanifold theory.
M f( i1 ! . i m ! im =0 i1 +···+im+n =L+1 t × 0 m,0 (h 1 i1 ) ... 1 i1 ! . i m ! m,0 (h m,0 (h 1 i1 m,0 (x)) m im ). . ) m,0 (h m im ) S (∂m+n )im . . (∂1S )i1 f (x + th)dt . 1 and thus has first derivatives satisfying (b). It also follows by induction that f is G∞ , so that (a) is established. Finally (d) follows from the fact that the classical Taylor expansion of the sum of two functions is the sum of the Taylor expansions of the functions, and the Taylor expansion of a product is the product of the Taylor expansions.
In this section a slightly restricted class of functions for finite-dimensional Grassmann algebras, known as GH ∞ functions, will be introduced; as the name suggests, GH ∞ functions occupy an intermediate position between G∞ and H ∞ functions. The basic idea of the definition is that a GH ∞ function has a Grassmann analytic expansion whose coefficient functions take their values in some subalgebra RS[L ] of RS[L] . ) Before defining these new functions the necessary subalgebras will be defined.