13. VECTOR-BUNDLE VALUED FIELD THEORIES 79 (3) For all L ∈ (0, ∞) and all x ∈ X, the limit lim ε→0 W(I,K) ⎛ ⎝P (ε, L),I − (i,k)≤(I,K) i ICT (i,k) (ε)⎠ ⎞ x exists in the topological vector space Hom(E ⊗K , R) ⊗ Ax. If we can prove these three properties, we can continue the induction. The third property is immediate: it follows from the small ε asymptotic expansion of Theorem 13.4.4. For the first property, observe that ∂ ∂L Sing ε W(I,K) ⎛ ⎝P (ε, L),I − (r,s) (I,K) r ICT (r,s) (ε)⎠ ⎞ = Singε ∂ ∂L W(I,K) ⎛ ⎝P (ε, L),I − (r,s) (I,K) r ICT (r,s) (ε)⎠ ⎞ . This follows from the fact that the small ε asymptotic expansion proved in Theorem 13.4.4 commutes with taking L derivatives. Thus, to show that ICT (I,K) is independent of L, it suﬃces to show that, for all L, all x ∈ X, and all e ∈ E , ∂ ∂L W (I,K) ⎛ ⎝P (ε, L),I − (r,s) (I,K) r ICT (r,s) (ε)⎠ ⎞ x (e) ∈ Ax has an ε → 0 limit. This is immediate by induction, using the renormaliza- tion group equation. The small L asymptotic expansion in Theorem 13.4.4 now implies that ICT (I,K) (ε) is local. Thus, ICT (I,K) (ε) satisfies all the required properties, and we can continue our induction, to construct all the counterterms. So far, we have shown how to construct the effective interactions I[L] = lim ε→0 W ⎛ ⎝P (ε, L),I − (r,s) r ICT (r,s) (ε)⎠ ⎞ . These effective interactions satisfy the renormalization group equation. It is immediate from Theorem 13.4.4 that the I[L] satisfy the locality axiom, so that they define a theory. Now, we need to prove the converse. This is again an inductive argument. Suppose we have a theory, given by effective interactions I[L] ∈ O+(E , A )[[ ]].

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