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Page 249
Nelson Dunford, Jacob T. Schwartz. PROOF . The identity | x + y2 + | x — y | 2 = 2 | x | 2 + 2 | y | 2 , x , yeh , called the parallelogram identity , follows immediately from the axioms . If d = inf x - k the preceding identity shows ...
Nelson Dunford, Jacob T. Schwartz. PROOF . The identity | x + y2 + | x — y | 2 = 2 | x | 2 + 2 | y | 2 , x , yeh , called the parallelogram identity , follows immediately from the axioms . If d = inf x - k the preceding identity shows ...
Page 479
... identity on X * . Thus ( T * ) - 1 exists , is in B ( X * , Y * ) , and equals ( T - 1 ) * . Conversely , if ( T ... identity 2 ( Tx , y ) = ( x , T VI.2.6 479 ADJOINTS.
... identity on X * . Thus ( T * ) - 1 exists , is in B ( X * , Y * ) , and equals ( T - 1 ) * . Conversely , if ( T ... identity 2 ( Tx , y ) = ( x , T VI.2.6 479 ADJOINTS.
Page 661
... identity → 0 . ( * ) Tn n 1 = A ( n ) A ( n - 1 ) n n shows that { T / n } is bounded and hence , by II.1.18 , the set of x for which Tx / n → 0 is a closed linear manifold . Thus X is a closed linear manifold and , since a continuous ...
... identity → 0 . ( * ) Tn n 1 = A ( n ) A ( n - 1 ) n n shows that { T / n } is bounded and hence , by II.1.18 , the set of x for which Tx / n → 0 is a closed linear manifold . Thus X is a closed linear manifold and , since a continuous ...
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