## Linear Operators: Spectral theory |

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Page 1105

We now pause to sharpen another of the

continuity of the product TS which was noted in the paragraph following Lemma 9

, the continuity of the norm function which follows from the triangle

Lemma ...

We now pause to sharpen another of the

**inequalities**of Lemma 9 . ... thecontinuity of the product TS which was noted in the paragraph following Lemma 9

, the continuity of the norm function which follows from the triangle

**inequality**ofLemma ...

Page 1774

The above

follows from the postulates for § that the Schwarz

is zero. Hence suppose that x 0 ^ y. For an arbitrary complex number a 0 ^ (x+ay,

...

The above

**inequality**, known as the Schwarz**inequality**, will be proved first. Itfollows from the postulates for § that the Schwarz

**inequality**is valid if either x or yis zero. Hence suppose that x 0 ^ y. For an arbitrary complex number a 0 ^ (x+ay,

...

Page 1774

The above

follows from the postulates for H that the Schwarz

is zero . Hence suppose that # # 0 # y . For an arbitrary complex number a 0 = ( x

...

The above

**inequality**, known as the Schwarz**inequality**, will be proved first . Itfollows from the postulates for H that the Schwarz

**inequality**is valid if either x or yis zero . Hence suppose that # # 0 # y . For an arbitrary complex number a 0 = ( x

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero