Linear Operators: General theory |
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Page 10
... containing p . A neighborhood of the set A is an open set containing 4. If A is a subset of X , then a point p is a limit point , or a point of accumulation , of A provided every neighborhood of p contains at least one point q #p , with ...
... containing p . A neighborhood of the set A is an open set containing 4. If A is a subset of X , then a point p is a limit point , or a point of accumulation , of A provided every neighborhood of p contains at least one point q #p , with ...
Page 20
... contains a non - void open set . Thus A1 ‡ X , and A ' is open , and contains a sphere S1 = S ( P1 , & 1 ) with 0 < & < 1/2 . By assumption , the set A does not contain the open set S ( P1 , & 1 / 2 ) ; hence the non - void open set 42S ...
... contains a non - void open set . Thus A1 ‡ X , and A ' is open , and contains a sphere S1 = S ( P1 , & 1 ) with 0 < & < 1/2 . By assumption , the set A does not contain the open set S ( P1 , & 1 / 2 ) ; hence the non - void open set 42S ...
Page 309
... contains no atoms having measure greater than ɛ . It will now be shown that every measurable subset B of A contains a set F with 0 < μ ( F ) ≤ ɛ . Sup- pose , on the contrary , that some such set B contains no set of Σ with positive ...
... contains no atoms having measure greater than ɛ . It will now be shown that every measurable subset B of A contains a set F with 0 < μ ( F ) ≤ ɛ . Sup- pose , on the contrary , that some such set B contains no set of Σ with positive ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
Copyright | |
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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