1. DISTRIBUTIONS We recall in this section a few facts concerning the cross-section spaces of vector bundles and state a theorem which will play an important role in Section 8. Its proof will be given in Section 10. 1.1. NOTATIONS. Let M be a manifold and V a smooth real vector bundle of finite rank over M. We denote by r(V) the space of all the C°° cross-sections of V. Equipped with the C°°-topology, r(V) is a complete nuclear space. Put r c (V) ~ {s T(V) supp s is compact}. Equipped with the usual inductive limit topology, also r (V) is a complete nuclear space. In the case V = M X R, we denote FM = r(V) and FJJ= r c (v)• F M is a topological algebra and T(W) is a topological FM-module in a natural way for any vector bundle W. Put 0'(V) - rc(V» ® AnT') ' and if'(V) = r(Vf ® Anx') ' (n = dim M, x - TM). Here Vf is the dual bundle of V and U' is the dual space with the strong topology of a topological vector space U. Note & (V) a^f(M) 8L r(V), ^•(V) aj^'(M) «L r(V), where 3X (M) »^'(MxR) and if1(M) ^»(MxR), considered as FM-modules naturally. Now, let V be an inductive limit bundle, that is, V = ind lim V., where V, c v C ... is a countable inductive system of vector bundles of finite rank. We put r(V) - ind lim r(V.), r (V) - ind lim r (V.), j c c j ®* (V) = ind lim^f(V.), and &' (V) - ind limi^1(V.). Endowed with the inductive limit topologies, they are complete nuclear spaces. 1.2. KERNEL THEOREMS. Let V^ and V- be inductive limit bundles over 7
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