## Linear Operators: General theory |

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Page 60

2\x\ ' 1 1 1 ' 1 F 1 This shows that (i)

continuity of T at 0; so (iv)

sup \Tx\ is finite, then for an arbitrary x ^ 0, 1*1 si This shows that (iii)

2\x\ ' 1 1 1 ' 1 F 1 This shows that (i)

**implies**(iv). Statement (iv) clearly**implies**thecontinuity of T at 0; so (iv)

**implies**(ii). This (i), (ii), and (iv) are equivalent. If M =sup \Tx\ is finite, then for an arbitrary x ^ 0, 1*1 si This shows that (iii)

**implies**(iv).Page 430

We observe first that each of the three conditions

metric topology, for x*(A ) is a bounded set of scalars for each x* e X* and we can

apply II. 3. 20. It is readily seen that (iii)

We observe first that each of the three conditions

**implies**that A is bounded in themetric topology, for x*(A ) is a bounded set of scalars for each x* e X* and we can

apply II. 3. 20. It is readily seen that (iii)

**implies**(ii). The other implications we ...Page 454

4. 2

then {&,□} converges, say, to qeK. If {A:^} is another such minimizing sequence

converging to q'eK, then IV. 4. 2

.

4. 2

**implies**that if {kt} is a sequence in A such that lim,..,.^ \p— kt\ =inf WK|p— k\then {&,□} converges, say, to qeK. If {A:^} is another such minimizing sequence

converging to q'eK, then IV. 4. 2

**implies**that {k^, k[, fc2, k'2, . . .} is also convergent.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero