Linear Operators: General theory |
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Page 22
... sequentially compact . Conversely , suppose that A is sequentially compact and closed . It will first be shown that A is separable . Let po be an arbitrary point of A , and let do sup o ( Po- p ) . The number do is finite , for if = PEA ...
... sequentially compact . Conversely , suppose that A is sequentially compact and closed . It will first be shown that A is separable . Let po be an arbitrary point of A , and let do sup o ( Po- p ) . The number do is finite , for if = PEA ...
Page 294
... sequentially compact then lim f ( s ) u ( ds ) = 0 , μ ( E ) → 0 E uniformly for fin K. If μ ( S ) < ∞ then conversely this condition is suffi- cient for a bounded set K to be weakly sequentially compact . PROOF . Let K be a weakly ...
... sequentially compact then lim f ( s ) u ( ds ) = 0 , μ ( E ) → 0 E uniformly for fin K. If μ ( S ) < ∞ then conversely this condition is suffi- cient for a bounded set K to be weakly sequentially compact . PROOF . Let K be a weakly ...
Page 314
... sequentially com- pact if and only if there exists a non - negative μ in ba ( S , E ) such that uniformly for λεΚ . lim λ ( E ) = 0 μ ( E ) → 0 PROOF . Let KC ba ( S , E ) be weakly sequentially compact and let V = UT be the isometric ...
... sequentially com- pact if and only if there exists a non - negative μ in ba ( S , E ) such that uniformly for λεΚ . lim λ ( E ) = 0 μ ( E ) → 0 PROOF . Let KC ba ( S , E ) be weakly sequentially compact and let V = UT be the isometric ...
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