## Linear Operators: General theory |

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Page 147

Q.E.D. 8 Lemma. // (S, Z, /a) is a measure space, a function f defined on S to X is

a null function if and only if it vanislies almost everywhere. If / is fi-integrable then

j Ef(s)fi(ds) = 0 for every E in E if and only if f

Q.E.D. 8 Lemma. // (S, Z, /a) is a measure space, a function f defined on S to X is

a null function if and only if it vanislies almost everywhere. If / is fi-integrable then

j Ef(s)fi(ds) = 0 for every E in E if and only if f

**vanishes**almost everywhere. Proof.Page 572

Thus, if TJ1 is the union of those spheres S(a.v e(a,)) which contain an infinite

number of points of a(T), then /

but a finite number of isolated points of a(T), which we suppose to be the points {

Xl, ...

Thus, if TJ1 is the union of those spheres S(a.v e(a,)) which contain an infinite

number of points of a(T), then /

**vanishes**identically on Uv Hence, V1 contains allbut a finite number of isolated points of a(T), which we suppose to be the points {

Xl, ...

Page 593

However, it is not obvious that the convergence of {fn(T)} can be deduced if /

possibility. 1 Theorem. Let f, f„ be in &{T), and let {f(T)f„(T)} converge to zero in the

uniform ...

However, it is not obvious that the convergence of {fn(T)} can be deduced if /

**vanishes**at certain points of a(T). The next theorem is concerned with thispossibility. 1 Theorem. Let f, f„ be in &{T), and let {f(T)f„(T)} converge to zero in the

uniform ...

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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