## Linear Operators: General theory |

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Page 245

Every linear operator on a

Proof. Let {bv . . ., bn} be a Hamel basis for the

space X so that every x in X has a unique representation in the form x = a^-j-.

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 linearspace X so that every x in X has a unique representation in the form x = a^-j-.

Page 246

Then the dimension of X** is finite, and. since X is equivalent to a subspaee of X

** (II.3.19), the dimension of X is finite. ... the same in

and from Corollary 3 that a set is compact if and only if it is bounded and closed.

Then the dimension of X** is finite, and. since X is equivalent to a subspaee of X

** (II.3.19), the dimension of X is finite. ... the same in

**finite dimensional**spacesand from Corollary 3 that a set is compact if and only if it is bounded and closed.

Page 556

Spectral Theory in a

denote a

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

Spectral Theory in a

**Finite Dimensional**Space Throughout this section, 3£ willdenote a

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

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional 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 Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact