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 = sup f ( s ) . 14. The space C ( S ) is defined for a topological space S ...
... 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 = sup f ( s ) . 14. The space C ( S ) is defined for a topological space S ...
Page 256
... scalar product ( iv ) ( [ x1 , . . . , xn ] , [ Y1 , • • • , Yn ] ) = n 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 ( iv ) ( [ x1 , . . . , xn ] , [ Y1 , • • • , Yn ] ) = n 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 of f with respect to u over E is an unambiguously defined element of X ; ( b ) if ƒ and g are scalar valued and μ - integrable , if x and ẞ are sca- lars , and if E € Σ , then Ε ↓ { aƒ ...
... scalar valued and μ - integrable , the integral of f with respect to u over E is an unambiguously defined element of X ; ( b ) if ƒ and g are scalar valued and μ - integrable , if x and ẞ are sca- lars , and if E € Σ , then Ε ↓ { aƒ ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
Copyright | |
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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 Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ