## Linear Operators, Part 1 |

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Page 245

Every linear operator on a

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

space X so that every x in X has a unique representation in the form x = abat ...

Every linear operator on a

**finite**dimensional normed linear space is continuous .Proof . Let { bı , ... , 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 = abat ...

Page 246

Then the dimension of X ** is

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

and X * have the same dimension . Q.E.D. In the preceding lemma it is ...

Then the 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 , Xand X * have the same dimension . Q.E.D. In the preceding lemma it is ...

Page 290

It is obvious that \ w * ) = [ gc , and so fæ * 1 = ' glcNow suppose that ( S , E , u ) is

o -

measure whose union is S. Using the theorem for L ( En ) = L ( En , E ( En ) , u ) ,

we ...

It is obvious that \ w * ) = [ gc , and so fæ * 1 = ' glcNow suppose that ( S , E , u ) is

o -

**finite**, and let En be an increasing sequence of measurable sets of**finite**measure whose union is S. Using the theorem for L ( En ) = L ( En , E ( En ) , u ) ,

we ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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