## Linear Operators: Spectral theory |

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Page 970

If X, denotes the characteristic function of the set e in 3, and if f is in L2(R), then X,

f is in L1(R) on L2(R) and f is the

sequence {X, f). Hence, by Theorem 9, rf is the

If X, denotes the characteristic function of the set e in 3, and if f is in L2(R), then X,

f is in L1(R) on L2(R) and f is the

**limit**in the norm of L2(R) of the generalizedsequence {X, f). Hence, by Theorem 9, rf is the

**limit**in the norm of L2(~40) of the ...Page 1124

If En, E are in 3% and p(Ea) increases to the

we have already proved that E, is an increasing sequence of projections and E, s

E. If Exe is the strong

If En, E are in 3% and p(Ea) increases to the

**limit**p(E), then it follows from whatwe have already proved that E, is an increasing sequence of projections and E, s

E. If Exe is the strong

**limit**of En, then E. s. E and p(Ex) = p(E). Thus, it follows as ...Page 1699

F. is the

g,(a) = 0 for a in Ce—L, it follows from Definition 3.15 that per's is the

norm of Ho(Ce) of the sequence {g}} of elements of Co(Ce). Hence, by Lemma ...

F. is the

**limit**in the norm of H*)(L) of a sequence {g}} of functions in Co(L). Puttingg,(a) = 0 for a in Ce—L, it follows from Definition 3.15 that per's is the

**limit**in thenorm of Ho(Ce) of the sequence {g}} of elements of Co(Ce). Hence, by Lemma ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 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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additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero