Linear Operators: General theory |
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Page 294
... weakly sequentially compact , a subsequence of { f } con- verges weakly , and it may be assumed without loss of generality that { f } itself converges weakly . By Theorem 7 , { SEfn ( s ) μ ( ds ) } converges for each EeΣ . Let ^ ‚ ( E ) ...
... weakly sequentially compact , a subsequence of { f } con- verges weakly , and it may be assumed without loss of generality that { f } itself converges weakly . By Theorem 7 , { SEfn ( s ) μ ( ds ) } converges for each EeΣ . Let ^ ‚ ( E ) ...
Page 314
... weakly 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 ...
... weakly 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 ...
Page 345
... weakly sequentially compact if and only if it is bounded and the set { f ( P ) } , fe A , is quasi - equicontinuous . 37 Let D be a bounded domain . Show that a sequence of func- tions in 4 ( D ) is a weak Cauchy sequence ( a sequence ...
... weakly sequentially compact if and only if it is bounded and the set { f ( P ) } , fe A , is quasi - equicontinuous . 37 Let D be a bounded domain . Show that a sequence of func- tions in 4 ( D ) is a weak Cauchy sequence ( a sequence ...
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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