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ak(x analogous assumptions Banach space bounded coefficients bounded variation Cauchy's problem characteristic polynomial Consider converges defined denoted derivatives of order double differential operator double order dual dual space duality equivalence fact finishes the proof finite following lemma formula function f Hahn-Banach theorem Helly's theorem Hence hermitian homogeneous hyperbolic equations hyperbolic operators implies infinitely differentiable J+ f left side lemma Leray let f linear functional linear homeomorphism Lip0 Lip1 Lipp'q m+l-k neighborhood of SQ norm normal notation operator of order order m+1 problem for hyperbolic proves the lemma remains to prove Remark replaced respectively result right side satisfying scalar product second kind sense shows Stieltjes integral suffices to prove tends to zero term theorem 7.1 theory of Cauchy's total order totally positive uniformly unique solution vanishing in neighborhoods variable vector space weakly dense write