## Linear Operators: General theory |

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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 p and q are distinct points of a locally convex linear topological space X , there

is a continuous

T2 ) .

If p and q are distinct points of a locally convex linear topological space X , there

is a continuous

**linear functional**f ... T , ) have the same continuous**linear****functionals**, then a convex set is closed in ( X , 7ı ) if and only if it is closed in ( X ,T2 ) .

Page 421

Let X be a linear space , and let l be a total subspace of X + . Then the

functionals in T ' . The proof of Theorem 9 will be based on the following lemma .

Let X be a linear space , and let l be a total subspace of X + . Then the

**linear****functionals**on X which are continuous in the r ' topology are precisely thefunctionals in T ' . The proof of Theorem 9 will be based on the following lemma .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Common terms and phrases

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