Linear Operators: General theory |
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Page 240
... scalar function f on S is E - measurable if f - 1 ( A ) for every Borel set A in the range of f . It is clear that ... scalar functions on S. The norm is given by = sup | f ( s ) . SES 14. The space C ( S ) is defined for a topological ...
... scalar function f on S is E - measurable if f - 1 ( A ) for every Borel set A in the range of f . It is clear that ... scalar functions on S. The norm is given by = sup | f ( s ) . SES 14. The space C ( S ) is defined for a topological ...
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
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
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 closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism 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 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ