## Linear Operators, Part 2 |

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Page 890

One class of

T ) has already been defined is the ... ( T ) determined by T provided that we

require the

elements .

One class of

**scalar**functions f , other than polynomials , for which the operator f (T ) has already been defined is the ... ( T ) determined by T provided that we

require the

**scalar**function g ( 2 ) = 2 and the operator T be correspondingelements .

Page 1178

Then x maps

from Plancherel ' s theorem that X is a bounded mapping of the space L , of

valued ...

Then x maps

**scalar**- valued functions into functions with values in lg . It is plainfrom Plancherel ' s theorem that X is a bounded mapping of the space L , of

**scalar**- valued functions into the space L2 ( 12 ) of square - integrable vector -valued ...

Page 1773

... and if limen , m + 00 ( an — Xm , & n — Xm ) = 0 , then there is an æ in H with

lim , ( xn — x , xn — X ) = 0 . The function ( : , • ) is called the

product in H and ( x , y ) is called the

in H ...

... and if limen , m + 00 ( an — Xm , & n — Xm ) = 0 , then there is an æ in H with

lim , ( xn — x , xn — X ) = 0 . The function ( : , • ) is called the

**scalar**or innerproduct in H and ( x , y ) is called the

**scalar**or inner product of x and y . The normin H ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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additive adjoint operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function give given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem 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 vanishes vector zero