Linear Operators: General theory |
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Page 372
... proved that any finite dimensional topolo- gical linear space is equivalent to an Euclidean space . This , in par ... proved in 1773. For a finite sum , it was proved in 1821 by Cauchy [ 2 ; p . 373 ] . It was proved for integrals in ...
... proved that any finite dimensional topolo- gical linear space is equivalent to an Euclidean space . This , in par ... proved in 1773. For a finite sum , it was proved in 1821 by Cauchy [ 2 ; p . 373 ] . It was proved for integrals in ...
Page 373
... proved for closed linear manifolds in L2 [ 0 , 1 ] by E. Fischer [ 2 ] . The fact that a linear manifold which is not dense in the entire space has a non - zero orthogonal complement ( proved in 4.4 ) was proved without the assumption ...
... proved for closed linear manifolds in L2 [ 0 , 1 ] by E. Fischer [ 2 ] . The fact that a linear manifold which is not dense in the entire space has a non - zero orthogonal complement ( proved in 4.4 ) was proved without the assumption ...
Page 385
... proved independently by Cech [ 1 ] only slightly later . ( See also Stone [ 5 ] for an elementary treatment . ) Lemma 6.25 was proved in Stone [ 1 ; p . 465 ] —extensions of this re- sult are also found in Hewitt [ 5 ] and Kaplansky ...
... proved independently by Cech [ 1 ] only slightly later . ( See also Stone [ 5 ] for an elementary treatment . ) Lemma 6.25 was proved in Stone [ 1 ; p . 465 ] —extensions of this re- sult are also found in Hewitt [ 5 ] and Kaplansky ...
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