Linear Operators: General theory |
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Page 240
... scalar function f on S is E - measurable if f - 1 ( A ) E for every Borel set A in the range of f . It is clear that ... scalar functions on S. The norm is given by 14 . If = sup f ( s ) . SES The space C ( S ) is defined for a ...
... scalar function f on S is E - measurable if f - 1 ( A ) E for every Borel set A in the range of f . It is clear that ... scalar functions on S. The norm is given by 14 . If = sup f ( s ) . SES The space C ( S ) is defined for a ...
Page 256
... scalar product n ( iv ) ( [ x1 , ... , xn ] , [ Y1 , · · · , Yn ] ) = Σ ( xi , Yi ) i , i = 1 where ( · , · ) ; is the scalar product in X. Thus the norm in a direct sum of Hilbert spaces is always given by ( iii ) . To summarize , we ...
... scalar product n ( iv ) ( [ x1 , ... , xn ] , [ Y1 , · · · , Yn ] ) = Σ ( xi , Yi ) i , i = 1 where ( · , · ) ; is the scalar product in X. Thus the norm in a direct sum of Hilbert spaces is always given by ( iii ) . To summarize , we ...
Page 323
... scalar valued and μ - integrable , the integral off with respect to μ over E is an unambiguously defined element of X ; ( b ) if ƒ and g are scalar valued and μ - integrable , if a and ẞ are sca- lars , and if Eε Σ , then √2 { aƒ ( s ) ...
... scalar valued and μ - integrable , the integral off with respect to μ over E is an unambiguously defined element of X ; ( b ) if ƒ and g are scalar valued and μ - integrable , if a and ẞ are sca- lars , and if Eε Σ , then √2 { aƒ ( s ) ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ