## Linear Operators: General theory |

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Page 59

A linear space 3E is a

corresponds a real number \x\ called the norm of x which satisfies the conditions:

(i) |0| = 0; \x\ > 0, x ^ 0; (ii) \x+y\ ^ + x,yel; (iii) \olx\ = \ct\\x\, x e 2i. The zero element

...

A linear space 3E is a

**normed linear space**, or a normed space, if to each x € 36corresponds a real number \x\ called the norm of x which satisfies the conditions:

(i) |0| = 0; \x\ > 0, x ^ 0; (ii) \x+y\ ^ + x,yel; (iii) \olx\ = \ct\\x\, x e 2i. The zero element

...

Page 65

For every x ^ 0 in a

= \x\. Proof. Apply Lemma 12 with 2J = 0. The x* required in the present corollary

may then be defined as \x\ times the x* whose existence is established in ...

For every x ^ 0 in a

**normed linear space**X, there is an x* e X* with \x*\ = 1 and x*x= \x\. Proof. Apply Lemma 12 with 2J = 0. The x* required in the present corollary

may then be defined as \x\ times the x* whose existence is established in ...

Page 245

Every linear operator on a finite dimensional

Proof. Let {bv . . ., bn} be a Hamel basis for the finite dimensional

Every linear operator on a finite dimensional

**normed linear space**is continuous.Proof. Let {bv . . ., bn} be a Hamel basis for the finite dimensional

**normed linear****space**X so that every x in X has a unique representation in the form x = a^-j-.### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact