Linear Operators: General theory |
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Page 240
... scalar function f on S is E - measurable if f − 1 ( A ) € Σ for every Borel set A in the range of f . It is clear ... scalar functions on S. The norm is given by If = sup f ( s ) . 14. The space C ( S ) is defined for a topological ...
... scalar function f on S is E - measurable if f − 1 ( A ) € Σ for every Borel set A in the range of f . It is clear ... scalar functions on S. The norm is given by If = sup f ( s ) . 14. The space C ( S ) is defined for a topological ...
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 u - integrable , if a and ẞ are sca- lars , and if E e 2 , then √ { xt ( 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 u - integrable , if a and ẞ are sca- lars , and if E e 2 , then √ { xt ( 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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ