## Linear Operators: General theory |

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Page 244

prototype of all n-dimensional normed linear spaces, and hence it should be

observed first that En is a B-space. The Minkowski inequality (III.8.3) shows E" to

be a ...

**Finite Dimensional**Spaces The space En, as will be seen presently, is theprototype of all n-dimensional normed linear spaces, and hence it should be

observed first that En is a B-space. The Minkowski inequality (III.8.3) shows E" to

be a ...

Page 245

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

on a

be a Hamel basis for the

An n-dimensional B-space is equivalent to En. 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 ...Page 246

Thus the

the

3.19), 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.3.19), the

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

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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 closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero