By Herbert Seifert, William Threlfall
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Extra info for A Textbook of Topology; Topology of 3-Dimensional Fibered Spaces
Im 1 im ∂1i1 . . ∂m f( i ! . i ! 1 m =0 m,n (x)) × s(x1 )i1 . . s(xm )im . 11) Given f in C ∞ (V, RS[L] ), the function f is called the Grassmann analytic continuation of f . This definition can be applied to a C ∞ function whose codomain is a subset of RS[L] , and to a set with a natural inclusion in RS[L] ; a case which occurs frequently is the continuation of a real-valued C ∞ function. function on ∞ Rm to Rm,n and that their derivatives correspond in a natural way. 3 Suppose that U is an open subset of Rm,n S[L] , and that ∞ f and g are functions in C ( m,n (U ), RS[L] ), the space of C ∞ maps of m,n (U ) into RS[L] regarded simply as a Banach space over the reals.
1 with (x; ξ) replaced by 1 ( (x )1, . . , (xm )1; 0 . . , 0) and (h; η) by (s(x1 ), . . , s(xm ); θ1 , . . 14), with the coefficient functions fµ : m,n (U ) → RS[L] defined by fµ (t1 , . . , tm ) = ∂µOk . . ∂µO1f (t1 1, . . , tm 1). 14) for some functions fµ in C ∞ ( U, RS[L] ). This corollary provides an alternative characterisation of G∞ functions which could equally well have been used as a definition, without recourse to any topology other than the DeWitt topology on the Grassmann algebra.
2 Let U be open in Rm,n S . 18) µ∈Mn for each (x; ξ) in U . This expansion is called the Grassmann analytic expansion of f and the functions fµ are called the Grassmann analytic coefficients of f . 3 Suppose that f is in G∞ (U ) with Grassmann expansion coefficients fµ . Then for i = 1, . . , m the derivative ∂iE f is defined to be the function in G∞ (U ) with Grassmann expansion coefficients ∂i fµ April 25, 2007 9:45 WSPC/Book Trim Size for 9in x 6in RogersSupermanifolds Supermanifolds: Theory and applications 40 while for j = 1, .
A Textbook of Topology; Topology of 3-Dimensional Fibered Spaces by Herbert Seifert, William Threlfall