Linear Operators: General theory |
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Page 89
... linear spaces which are not complete . In such cases , the following theorem is often ... linear space satisfying properties ( i ) and ( ii ) of Definition 1.10 . Then X is isomorphic and isometric with a dense linear subspace of an F - space ...
... linear spaces which are not complete . In such cases , the following theorem is often ... linear space satisfying properties ( i ) and ( ii ) of Definition 1.10 . Then X is isomorphic and isometric with a dense linear subspace of an F - space ...
Page 91
... linear metric space can be metrized to be an F - space . Further , a normed linear space is a B - space provided it is complete under some equivalent metric . See also van Dantzig [ 1 ] , [ 2 ] . Norms in linear spaces . We have seen ...
... linear metric space can be metrized to be an F - space . Further , a normed linear space is a B - space provided it is complete under some equivalent metric . See also van Dantzig [ 1 ] , [ 2 ] . Norms in linear spaces . We have seen ...
Page 239
... space is the linear space of all ordered n - tuples [ 1 , ... , an ] of scalars 1 ,. an with the norm = sup [ x ] . 1≤i≤n 4. The space l , is defined for 1 ≤ p < ∞ as the linear ... linear space of all IV.2.3 239 A LIST OF SPECIAL SPACES.
... space is the linear space of all ordered n - tuples [ 1 , ... , an ] of scalars 1 ,. an with the norm = sup [ x ] . 1≤i≤n 4. The space l , is defined for 1 ≤ p < ∞ as the linear ... linear space of all IV.2.3 239 A LIST OF SPECIAL SPACES.
Contents
Preliminary Concepts | 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 ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ