Linear Operators: General theory |
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Page 46
... C ( i , ... , ipi i j ) , then the Laplace expansion of det ( a ,, ) in terms of the elements of the i ,. ith rows is given by the formula .... det ( as ) = Σ B ( i . ipi i ip ) × C ( i ,. · • .... ipi iv ... , İp ) , i1 , ... , jp ...
... C ( i , ... , ipi i j ) , then the Laplace expansion of det ( a ,, ) in terms of the elements of the i ,. ith rows is given by the formula .... det ( as ) = Σ B ( i . ipi i ip ) × C ( i ,. · • .... ipi iv ... , İp ) , i1 , ... , jp ...
Page 344
... C ( I ) determines an isometric isomorphism between C ( I ) * and NBV ( I ) . Show that if { gn } is a sequence of functions in NBV ( I ) , we have lim if ( s ) dgn ( s ) = √1f ( s ) dgo ( s ) for all fin C ( I ) if and only if ( i ) v ...
... C ( I ) determines an isometric isomorphism between C ( I ) * and NBV ( I ) . Show that if { gn } is a sequence of functions in NBV ( I ) , we have lim if ( s ) dgn ( s ) = √1f ( s ) dgo ( s ) for all fin C ( I ) if and only if ( i ) v ...
Page 558
... ( c ) I = Σ E ( 2 ) . λεσ ( Τ ) X ; Let { 21 , . . . , λ % } be an enumeration of o ( T ) , and let X1 = E ( 2 ) X- It follows from ( b ) and ( c ) of Theorem 6 that X = X1 ... X. X1 ® . . . ® X1⁄2 • 1 i Moreover , since TE ( 2 ) E ( 2 ) ...
... ( c ) I = Σ E ( 2 ) . λεσ ( Τ ) X ; Let { 21 , . . . , λ % } be an enumeration of o ( T ) , and let X1 = E ( 2 ) X- It follows from ( b ) and ( c ) of Theorem 6 that X = X1 ... X. X1 ® . . . ® X1⁄2 • 1 i Moreover , since TE ( 2 ) E ( 2 ) ...
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