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Page 254
... Consequently { u } and { v } have the same cardinality . Q.E.D. 15 DEFINITION . The cardinality of an arbitrary orthonormal basis of a Hilbert space is its dimension . THEOREM . Two Hilbert spaces are isometrically isomorphic if and ...
... Consequently { u } and { v } have the same cardinality . Q.E.D. 15 DEFINITION . The cardinality of an arbitrary orthonormal basis of a Hilbert space is its dimension . THEOREM . Two Hilbert spaces are isometrically isomorphic if and ...
Page 451
... Consequently , letting Zn , i , i we have Z , - x | x € X , j { f ( x + Yn / j ) −2f ( x ) + k ( x— ¥ n / j ) } < { x \ x € X. - U = 1 1 Zij . If we put Zn , i - U - 1 Zn , i , j , then i = 1 = n = 1 022-1 Zn 00 - n = 1 11 , is dense i ...
... Consequently , letting Zn , i , i we have Z , - x | x € X , j { f ( x + Yn / j ) −2f ( x ) + k ( x— ¥ n / j ) } < { x \ x € X. - U = 1 1 Zij . If we put Zn , i - U - 1 Zn , i , j , then i = 1 = n = 1 022-1 Zn 00 - n = 1 11 , is dense i ...
Page 637
... Consequently , ( i ) ,. . . , ( v ) are proved inductively for all n . We now obtain an estimate for the series Σox ( n ) ( t ) which majorizes o Sn ( t ) . N = By Lemma 20 ( c ) , for each w > wo there exists a constant M∞ such that y ...
... Consequently , ( i ) ,. . . , ( v ) are proved inductively for all n . We now obtain an estimate for the series Σox ( n ) ( t ) which majorizes o Sn ( t ) . N = By Lemma 20 ( c ) , for each w > wo there exists a constant M∞ such that y ...
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