By Hart W. L.
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Additional info for Differential Equations and Implicit Functions in Infinitely Many Variables (1916)(en)(5s)
Let V = (V1 , . . , Vs ) and W = (W1 , . . , Wt ) be two direct sums of V . i) If each Vi is a subconnection, we say that V is a direct sum of subconnections of V . ii) We say that V is finer than W , and we write V ≺ W , if for any 1 exists 1 j t such that Vi ⊂ Wj . i s there iii) We say that V and W are compatible if there exists a direct sum Z, which is finer than V and W . iv) If V and W are compatible, by the intersection of V and W we mean the largest direct sum of V such that V ∧ W ≺ V and V ∧ W ≺ W .
Let - equivalence up to order T relation ∼ as follows. T i) u ∼ u˜ if u − u˜ ∈ zT +1 T ii) ∼ ˜ if T for vectors u and u˜ in V . − ˜ ∈ zT +1 ∗ ⊗O iii) ∇ ∼ ∇˜ if ∇θ − ∇˜ θ ∈ zT +1 T ⊗O for maps ∗ and ˜ ∈ EndK V can V ⊗K V ∗ . for connections ∇ and ∇˜ on V . iv) Let (ε) and (˜ε ) be bases of V . Then (ε) ∼ (˜ε ) if εi ∼ ε˜ i for all i = 1, . . , n. 2. With the given notation, one has i) (ε) ∼ (˜ε) if and only if P − P˜ ∈ zT +1 Mn (O), where P = P(e),(ε) and P˜ = P(e),(˜ε) T for any basis (e) of .
I=1 Hence, the collection Z˜ = (Wj )1 i m,1 j ti is a direct sum of V . The direct sum Z˜ is finer than Z by construction. Let us prove that it is also finer than W . Let E = /z and E = F1 ⊕ · · · ⊕ Ft be the sum of E into the characteristic spaces of f = ∇ . For (i) (i) any 1 i m, let i /z i = Gi and ∇i i = fi . Let Gi = F1 ⊕ · · · ⊕ Fti be the (i) direct sum of the characteristic spaces of fi , and λj the corresponding eigenvalue. (i) (i) Every Fj is contained in exactly one Fk . Hence we have t i=1 Pfi (X) characteristic polynomial Pf (X) satisfies of Pf (X) = (i) Fj (i) λj =λk (i) Fj ⊂ Fk .