## Linear Operators: General theory |

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Page 243

some of the concrete spaces defined above satisfy these axioms , and hence are

special cases of abstract

**Hilbert space**has been defined by a set of abstract axioms . It is noteworthy thatsome of the concrete spaces defined above satisfy these axioms , and hence are

special cases of abstract

**Hilbert space**. Thus , for instance , the n - dimensional ...Page 256

Whenever the direct sum of normed linear spaces is used as a normed space ,

the norm will be explicitly mentioned . If , however , each of the spaces X1 , . . . ,

Xn are

explicit ...

Whenever the direct sum of normed linear spaces is used as a normed space ,

the norm will be explicitly mentioned . If , however , each of the spaces X1 , . . . ,

Xn are

**Hilbert spaces**then it will always be understood , sometimes withoutexplicit ...

Page 838

space of, definition, IV.2.24 (242) properties, IV.15 Annihilator of a set, II.4.17 (72)

Arzela theorem, on continuity of limit ... (See Hamel base) orthogonal and

orthonormal bases in

(252) ...

space of, definition, IV.2.24 (242) properties, IV.15 Annihilator of a set, II.4.17 (72)

Arzela theorem, on continuity of limit ... (See Hamel base) orthogonal and

orthonormal bases in

**Hilbert space**, definition, IV.4.11 (252) existence of, IV.4.12(252) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero