## Linear Operators: General theory |

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Page 20

This notion is written symbolically a„ -> o, or lim^o, an = a. A sequence {a„} is said

to be convergent if a„ -v a for some a. A sequence {an} in a metric space is a

...

This notion is written symbolically a„ -> o, or lim^o, an = a. A sequence {a„} is said

to be convergent if a„ -v a for some a. A sequence {an} in a metric space is a

**Cauchy sequence**if limm p(am, «„) = 0. If every**Cauchy sequence**is convergent,...

Page 68

which converges weakly to a point in S. Every sequence {xn} such that {.r*a'M} is

a

which converges weakly to a point in S. Every sequence {xn} such that {.r*a'M} is

a

**Cauchy sequence**of scalars for each x* e S* is called a weak**Cauchy****sequence**. The space 3c is said to be weakly complete if every weak**Cauchy****sequence**...Page 122

Let 1 p < oo; let {/„} be a sequence of functions in L„(S, E, fi, S) and let f be a

function on S to X. Then f is in Lv and \in~f\v converges to ... Assuming (i), (ii), and

(iii) let g„= |/„(-)|', g=|/(')|p- We will first show that {g„} is a

.

Let 1 p < oo; let {/„} be a sequence of functions in L„(S, E, fi, S) and let f be a

function on S to X. Then f is in Lv and \in~f\v converges to ... Assuming (i), (ii), and

(iii) let g„= |/„(-)|', g=|/(')|p- We will first show that {g„} is a

**Cauchy sequence**in L^S).

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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 closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero