Linear Operators: General theory |
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Page 412
... linear functional on the vector space X , and let be a subspace of X. If f ( 9 ) is not the whole field of scalars , then f ( Y ) 0 . PROOF . Suppose that there exists a yeY with f ( y ) ‡ 0. Then f ( xy / f ( y ) ) a , so that every ...
... linear functional on the vector space X , and let be a subspace of X. If f ( 9 ) is not the whole field of scalars , then f ( Y ) 0 . PROOF . Suppose that there exists a yeY with f ( y ) ‡ 0. Then f ( xy / f ( y ) ) a , so that every ...
Page 418
... linear functional separates K and p . 13 COROLLARY . If p and q are distinct points of a locally convex linear topological space X , there is a continuous linear functional ƒ de- fined on such that f ( p ) #f ( q ) . As a last corollary ...
... linear functional separates K and p . 13 COROLLARY . If p and q are distinct points of a locally convex linear topological space X , there is a continuous linear functional ƒ de- fined on such that f ( p ) #f ( q ) . As a last corollary ...
Page 452
... linear topological space X , and K X , then there exists a non- zero continuous linear functional tangent to K at p . If A is a subset of X , and p is in A , then there exists a non - zero continuous linear functional tangent to A at p ...
... linear topological space X , and K X , then there exists a non- zero continuous linear functional tangent to K at p . If A is a subset of X , and p is in A , then there exists a non - zero continuous linear functional tangent to A at p ...
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