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Page 151
... Consequently , lim sup | − = lim sup | \ . ( s ) —m ( s ) | po ( u , ds ) + 2g / n m , n∞ m , n∞o lim sup [ { \ fn ( 8 ) — † ( 8 ) \ " v ( μ , ds ) } ' 1o m , n∞ + { { £ ̧ \ † m ( 8 ) − † ( 8 ) \ P v ( μ , ds ) } ) 11 ] + 2ɛ1 » 2ε1 ...
... Consequently , lim sup | − = lim sup | \ . ( s ) —m ( s ) | po ( u , ds ) + 2g / n m , n∞ m , n∞o lim sup [ { \ fn ( 8 ) — † ( 8 ) \ " v ( μ , ds ) } ' 1o m , n∞ + { { £ ̧ \ † m ( 8 ) − † ( 8 ) \ P v ( μ , ds ) } ) 11 ] + 2ɛ1 » 2ε1 ...
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 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