Linear Operators: General theory |
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Page 253
... Lemma 4 , M = A1 → ( M = A1 ) , and so M = A1 . The desired conclusion now follows from Theorem 10. Q.E.D. 13 THEOREM . For an orthonormal set AC the following state- ments are equivalent : ( i ) the set A is complete ; ( ii ) the set ...
... Lemma 4 , M = A1 → ( M = A1 ) , and so M = A1 . The desired conclusion now follows from Theorem 10. Q.E.D. 13 THEOREM . For an orthonormal set AC the following state- ments are equivalent : ( i ) the set A is complete ; ( ii ) the set ...
Page 422
... Theorem 2.10 , there exists a non- zero linear П - continuous functional g , and a real constant c , such that Rg ( ) c . By Lemma 1.11 , g ( Hƒ ) = 0 ; i.e. , f ( x ) = 0 implies g ( x ) = 0 . It follows from Lemma 10 that g af for ...
... Theorem 2.10 , there exists a non- zero linear П - continuous functional g , and a real constant c , such that Rg ( ) c . By Lemma 1.11 , g ( Hƒ ) = 0 ; i.e. , f ( x ) = 0 implies g ( x ) = 0 . It follows from Lemma 10 that g af for ...
Page 485
... follows from Lemma 7 and Lemma I.5.7 that T * S * is compact in the X ** topology of X * . Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma 7 that T ** is continuous relative to the X ...
... follows from Lemma 7 and Lemma I.5.7 that T * S * is compact in the X ** topology of X * . Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma 7 that T ** is continuous relative to the X ...
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