Linear Operators: General theory |
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Page 106
... S are the functions in the closure TM ( S ) in F ( S ) of the μ - simple functions . If for every E in Σ with v ( u , E ) < ∞ , the product XEf of f with the characteristic function XE of E is totally measurable , the function f is ...
... S are the functions in the closure TM ( S ) in F ( S ) of the μ - simple functions . If for every E in Σ with v ( u , E ) < ∞ , the product XEf of f with the characteristic function XE of E is totally measurable , the function f is ...
Page 329
Nelson Dunford, Jacob T. Schwartz. 11. The Space TM ( S , Σ , μ ) We shall be concerned here with a set S , a σ - field Σ of its subsets , and a scalar valued countably additive set function μ on Σ . The symbol TM ( S , E ... TM ( S , 2 , μ )
Nelson Dunford, Jacob T. Schwartz. 11. The Space TM ( S , Σ , μ ) We shall be concerned here with a set S , a σ - field Σ of its subsets , and a scalar valued countably additive set function μ on Σ . The symbol TM ( S , E ... TM ( S , 2 , μ )
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Nelson Dunford, Jacob T. Schwartz. linear functional on TM ( S , E , μ ) which is not identically zero , then , since linear combinations of characteristic functions of intervals are dense in TM ( S , E , μ ) , there is a subinterval 4 ...
Nelson Dunford, Jacob T. Schwartz. linear functional on TM ( S , E , μ ) which is not identically zero , then , since linear combinations of characteristic functions of intervals are dense in TM ( S , E , μ ) , there is a subinterval 4 ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ