## Linear Operators, Part 1 |

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Page 412

10 LEMMA . The

if it 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 ...

10 LEMMA . The

**linear functional**f separates the subsets M and N of X if and onlyif it 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 ...

Page 418

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

is a continuous

tz ) .

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

is a continuous

**linear functional**... Tz ) have the same continuous**linear****functionals**, then a convex set is closed in ( X , 17 ) if and only if it is closed in ( X ,tz ) .

Page 421

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

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

10 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 l ' topology are precisely thefunctionals in . . The proof of Theorem 9 will be based on the following lemma .

10 LEMMA .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero