Linear Operators: General theory |
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Page 59
... linear space X is a normed linear space , or a normed space , if to each a ex corresponds a real number a called the norm of x which satisfies the conditions : = ( i ) 0 0 ; x > 0 , x 0 ; ( ii ) │x + y | ≤ │x + y \ , x , y € X ...
... linear space X is a normed linear space , or a normed space , if to each a ex corresponds a real number a called the norm of x which satisfies the conditions : = ( i ) 0 0 ; x > 0 , x 0 ; ( ii ) │x + y | ≤ │x + y \ , x , y € X ...
Page 91
... linear metric space can be metrized to be an F - space . Further , a normed linear space is a B - space provided it is complete under some equivalent metric . See also van Dantzig [ 1 ] , [ 2 ] . Norms in linear spaces . We have seen that ...
... linear metric space can be metrized to be an F - space . Further , a normed linear space is a B - space provided it is complete under some equivalent metric . See also van Dantzig [ 1 ] , [ 2 ] . Norms in linear spaces . We have seen that ...
Page 245
... space is equivalent to E " . 4 COROLLARY . Every linear operator on a finite dimensional normed linear space is continuous . = PROOF . Let { b1 , . . . , b1 } be a Hamel basis for the finite dimension- al normed linear space X so that ...
... space is equivalent to E " . 4 COROLLARY . Every linear operator on a finite dimensional normed linear space is continuous . = PROOF . Let { b1 , . . . , b1 } be a Hamel basis for the finite dimension- al normed linear space X so that ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ