## Linear Operators: General theory |

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Page 245

An n-dimensional B-space is equivalent to E". 4 Corollary. Every linear operator

on a

be a Hamel basis for the

...

An n-dimensional B-space is equivalent to E". 4 Corollary. Every linear operator

on a

**finite dimensional**normed linear space is continuous. Proof. Let {bv . . ., bn}be a Hamel basis for the

**finite dimensional**normed linear space X so that every x...

Page 246

Thus the

the

8.19), the

...

Thus the

**dimension**of X* is n. Conversely, let the**dimension**of X* be**finite**. Thenthe

**dimension**of X** is**finite**, and. since X is equivalent to a subspace of X** (II.8.19), the

**dimension**of X is**finite**. Hence, from the first part of this proof, X and X*...

Page 556

Spectra] Theory in a

denote a

aim is to study the algebraic and topological properties of T. This is achieved,

here ...

Spectra] Theory in a

**Finite Dimensional**Space Throughout this section, X willdenote a

**finite dimensional**complex B-space, and T a linear operator in B(£). Ouraim is to study the algebraic and topological properties of T. This is achieved,

here ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

44 other sections not shown

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### Common terms and phrases

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 convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set 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 Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero