## 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.3.3) shows En 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.3.3) shows En 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 w. 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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