## Linear Operators: General theory |

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Page 14

If X, Y, Z are topological spaces, and if / : X -> Y and g :Y -> Z are

the composite

complex number; a scalar

If X, Y, Z are topological spaces, and if / : X -> Y and g :Y -> Z are

**continuous**, thenthe composite

**function**fg is**continuous**. The term scalar will be used for a real orcomplex number; a scalar

**function**is a real or complex valued**function**.Page 274

Let S be a compact Hausdorff space and C(S) be the algebra of all complex

the unit e and contains, with f, its complex conjugate f defined by f(s) = f(s). Then

21 ...

Let S be a compact Hausdorff space and C(S) be the algebra of all complex

**continuous functions**on S. Let 21 be a closed subalgebra of C(S) which containsthe unit e and contains, with f, its complex conjugate f defined by f(s) = f(s). Then

21 ...

Page 381

10], Hewitt [8] and Edwards [1] consider functionals on the space of

sets. Arens [8] discussed certain sub-algebras of

10], Hewitt [8] and Edwards [1] consider functionals on the space of

**continuous****functions**on a locally compact Hausdorff space which vanish outside of compactsets. Arens [8] discussed certain sub-algebras of

**continuous functions**on a ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 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

a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero