Linear Operators: General theory |
From inside the book
Results 1-3 of 67
Page 488
... range , then the range of U is closed and consists of those vectors y in for which U * y * = 0 implies y * y = 0 . Y 1 U ( X ) , defined by * U is one - to - one . If 1 PROOF . Consider the map U1 from X to 3 U1 ( x ) = U ( x ) . Then ...
... range , then the range of U is closed and consists of those vectors y in for which U * y * = 0 implies y * y = 0 . Y 1 U ( X ) , defined by * U is one - to - one . If 1 PROOF . Consider the map U1 from X to 3 U1 ( x ) = U ( x ) . Then ...
Page 489
... range of U is also closed . It follows from the previous lemma that U1X Hence , U has a closed range . Q.E.D. UX -- 1 3 . 5 THEOREM . If U is in B ( X , Y ) and maps bounded closed sets onto closed sets , then U has a closed range ...
... range of U is also closed . It follows from the previous lemma that U1X Hence , U has a closed range . Q.E.D. UX -- 1 3 . 5 THEOREM . If U is in B ( X , Y ) and maps bounded closed sets onto closed sets , then U has a closed range ...
Page 513
... range of U is closed . ( ii ) The range of U is closed if there exists a constant K such that for any y in the range there exists a solution of y Tx such that x ≤Ky . ( iii ) U is one - to - one if the range of U * is dense in X ...
... range of U is closed . ( ii ) The range of U is closed if there exists a constant K such that for any y in the range there exists a solution of y Tx such that x ≤Ky . ( iii ) U is one - to - one if the range of U * is dense in X ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
80 other sections not shown
Other editions - View all
Common terms and phrases
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