## Linear Operators: General theory |

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Page 50

The subspace spanned by a set B in a linear space X will be denoted by sp ( B ) .

If X is a linear topological space , the closure of sp ( B ) , denoted by sp ( B ) , is

called the

The subspace spanned by a set B in a linear space X will be denoted by sp ( B ) .

If X is a linear topological space , the closure of sp ( B ) , denoted by sp ( B ) , is

called the

**closed linear manifold**determined by , or spanned by , B . From ...Page 249

Two manifolds M , N in H are orthogonal manifolds if ( M , N ) = 0 . We write x I y

to indicate that x ... The orthocomplement N of a

Two manifolds M , N in H are orthogonal manifolds if ( M , N ) = 0 . We write x I y

to indicate that x ... The orthocomplement N of a

**closed linear manifold**M in ø is a**closed linear manifold**complementary to M , i . e . H = MON . PROOF . It follows ...Page 252

Nelson Dunford, Jacob T. Schwartz. converges and is independent of the order in

which its non - zero terms are arranged . The operator E is the orthogonal

projection on the

Yn be ...

Nelson Dunford, Jacob T. Schwartz. converges and is independent of the order in

which its non - zero terms are arranged . The operator E is the orthogonal

projection on the

**closed linear manifold**determined by A . PROOF . Let Yı , . . . ,Yn be ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero