## Some Expansions in Vector Space |

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A B„_ AB„_ Abel's theorem absolutely convergent an(T analytic func analytic function Applying the Lemma clearly sufficient complex number continuous functions Corollary D(XT defined definies an analytic Definition degree n differential equation 6.3 Expansions B(T Fasc follows immediately Frechet differential functions B(T Hence homogeneous polynomial hypothesis I A _ inequalities Integrate 8.4 inverse given Journ least unity left hand inverse Lemma taking less than unity linear continuous operation linear vector space mB„_ modulus non-negative real numbers number system obtain oD(T polynomial of degree postulates power series precisely the coefficient Proof properties proved elsewhere R. S. MARTIN radii of analyticity radii of convergence radius Reccurent recurrence recursion formula 4.3 resolvent of XT respectively right hand inverse rotation satisfies sequence of elements sequence of non-negative similar argument Taking norms tend to zero Theorem 3.I Theorem 4.Ill Theorem 5.I theorem is proved tions transposed unique unit circle XT)J