## 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 X 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 421

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

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

10 LEMMA .

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

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

10 LEMMA .

Page 452

If A is a subset of X , and p is in A , then there exists a non - zero continuous

by A is not dense in X . PROOF . If q¢ K , then , by 2 . 12 we can find a functional ...

If A is a subset of X , and p is in A , then there exists a non - zero continuous

**linear functional**tangent to A at p if and only if the cone B with vertex p generatedby A is not dense in X . PROOF . If q¢ K , then , by 2 . 12 we can find a functional ...

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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