Linear Operators: General theory |
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Page 240
... scalar function f on S is E - measurable if f - 1 ( 4 ) € Σ for every Borel set A in the range of f . It is clear ... scalar functions on S. The norm is given by 14 . = sup f ( s ) . SES The space C ( S ) is defined for a topological ...
... scalar function f on S is E - measurable if f - 1 ( 4 ) € Σ for every Borel set A in the range of f . It is clear ... scalar functions on S. The norm is given by 14 . = sup f ( s ) . SES The space C ( S ) is defined for a topological ...
Page 256
... scalar product n ( iv ) ( x1 , ... , xn ] , [ Y1 , ... , Yn ] ) = Σ ( xj , 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 state the ...
... scalar product n ( iv ) ( x1 , ... , xn ] , [ Y1 , ... , Yn ] ) = Σ ( xj , 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 state the ...
Page 323
... scalar valued and u - integrable , the integral of f with respect to μ over E is an unambiguously defined element of X ; α ( b ) if ƒ and g are scalar valued and μ - integrable , if a and ẞ are sca- lars , and if E e Σ , then √g { at ...
... scalar valued and u - integrable , the integral of f with respect to μ over E is an unambiguously defined element of X ; α ( b ) if ƒ and g are scalar valued and μ - integrable , if a and ẞ are sca- lars , and if E e Σ , then √g { at ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ