## Linear Operators, Part 1 |

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Page 412

The

separates the subsets M - N and { 0 } of X. The proof is elementary , and is left to

the reader . In dealing with subspaces , it is often convenient to make use of the

following ...

The

**linear functional**f separates the subsets M and N of X if and only if itseparates the subsets M - N and { 0 } of X. The proof is elementary , and is left to

the reader . In dealing with subspaces , it is often convenient to make use of the

following ...

Page 418

If Ky , K , are disjoint closed , convex subsets of a locally convex linear

topological space X , and if K , is compact , then some non - zero continuous

convex subset ...

If Ky , K , are disjoint closed , convex subsets of a locally convex linear

topological space X , and if K , is compact , then some non - zero continuous

**linear functional**on X separates K , and Kg . 12 COROLLARY . I / K is a closedconvex subset ...

Page 421

Let X be a linear space , and let T be a total subspace of X * . Then the

functionals in l . The proof of Theorem 9 will be based on the following lemma .

10 LEMMA .

Let X be a linear space , and let T be a total subspace of X * . Then the

**linear****functionals**on X which are continuous in the I topology are precisely thefunctionals in l . The proof of Theorem 9 will be based on the following lemma .

10 LEMMA .

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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