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Let if e £ . (ι-τ)ν) = o . Since this is true for each y e £ , it follows that f = *jf. 3. 6), the differentiability of the function a was essential. Although examples can be found in which even singular distributions can be multiplied, it is not possible to define a natural multiplication on the set of distributions, which is associative and commutative (L. 117)· By 'natural1, we mean that the multiplication of a distribution by an infinitely differentiable function should be a special case of the new definition.

According t o Taylor's theorem, we can write ip(x)= Σ. 13) where Θ e L and D θ(θ) = 0 for la| < m. 12) that - ^ ΐ and D < \(°) = °for ,α| ^ m# |(f^ k )l < c i h y n = C sup sup ( l + | x | ) m |D a yv(x)| - * 0 lai^m i x i i l / k as k —► oo. On the other hand, for each k, f = η, f since ^, = 1 on an open neighbourhood of 0. e1k) = (1kf,e) = (f,e) which is independent of k. (f,^k) = 0 Thus for k s 1,2,... 14) a (-lì\0C a Writing e ss >. ' (f,x ), (7*1^) becomes " a ~~ ca(DaSfy) .

UL . As B - U, is closed, there is an open subset G of B n that B - U k c G c G c w. n Define Vfc = B - G. and, since B n - G is closed, Vfe c U k · are satisfied with j = k. Then Vfc is open, Vfc c B - G c Ufc Further Vfc U W = B n . Thus (l) and (2) If x 6. B n , there is a largest positive integer p such that xfcU x e νΛ U ... U V .. such n and so Thus {V\; j ss 1,2,... } is an open covering of B n . For each j, V. 1, there is a function h. in & with support in U. such that range h. (x) ss 1 for each x G V..

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