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Page 245
Nelson Dunford, Jacob T. Schwartz. 3 COROLLARY . An n - dimensional B - space is equivalent to E " . 4 COROLLARY . Every linear operator on a finite dimensional normed linear space is continuous . PROOF . Let { b1 , . . . , b , } be a ...
Nelson Dunford, Jacob T. Schwartz. 3 COROLLARY . An n - dimensional B - space is equivalent to E " . 4 COROLLARY . Every linear operator on a finite dimensional normed linear space is continuous . PROOF . Let { b1 , . . . , b , } be a ...
Page 246
... dimension . Q.E.D. In the preceding lemma it is assumed that the spaces are normed linear spaces , for there are examples ( see the preliminary discussion in Section IV.11 ) of infinite dimensional F - spaces with zero dimensional ...
... dimension . Q.E.D. In the preceding lemma it is assumed that the spaces are normed linear spaces , for there are examples ( see the preliminary discussion in Section IV.11 ) of infinite dimensional F - spaces with zero dimensional ...
Page 556
Nelson Dunford, Jacob T. Schwartz. 1. Spectral Theory in a Finite Dimensional Space Throughout this section , X will denote a finite dimensional com- plex B - space , and T a linear operator in B ( X ) . Our aim is to study the algebraic ...
Nelson Dunford, Jacob T. Schwartz. 1. Spectral Theory in a Finite Dimensional Space Throughout this section , X will denote a finite dimensional com- plex B - space , and T a linear operator in B ( X ) . Our aim is to study the algebraic ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ