Linear Operators: General theory |
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Page 89
... seen to be a topological linear space . If X and Y are B- ( or F- ) spaces , then X is a B- ( or an F- ) space under either of the norms │ [ x , y ] ] = max ( x , y ) , [ [ x , y ] = { xP + yp } 1 / P , 1 ≤ p < ∞ , and these norms ...
... seen to be a topological linear space . If X and Y are B- ( or F- ) spaces , then X is a B- ( or an F- ) space under either of the norms │ [ x , y ] ] = max ( x , y ) , [ [ x , y ] = { xP + yp } 1 / P , 1 ≤ p < ∞ , and these norms ...
Page 181
... seen that it will be sufficient to prove [ * ] in the case when g is a positive real function . It is also clear that [ * ] holds for 2 - integrable simple functions . As in the proof of Theorem 4 , there is an increasing sequence { g } ...
... seen that it will be sufficient to prove [ * ] in the case when g is a positive real function . It is also clear that [ * ] holds for 2 - integrable simple functions . As in the proof of Theorem 4 , there is an increasing sequence { g } ...
Page 254
... seen that vg is equivalent to vg and thus that υβ v is in V. Since { v } is a basis , the vector u , has an expansion of the form u ( u , v ) , so that u , is in the closed linear manifold de- termined by those υβ with ( u , v ) 0 ...
... seen that vg is equivalent to vg and thus that υβ v is in V. Since { v } is a basis , the vector u , has an expansion of the form u ( u , v ) , so that u , is in the closed linear manifold de- termined by those υβ with ( u , v ) 0 ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ