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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 Hamel ...
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 Hamel ...
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
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ