## Linear Operators: Spectral theory |

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Page 1775

Two

to indicate that x and y are orthogonal , and M 1 N to indicate that M and N are

orthogonal . The orthocomplement of a set A CH is the set { x | ( x , A ) = 0 } .

Two

**manifolds**M , N in H are orthogonal**manifolds**if ( M , N ) = 0 . We write x l yto indicate that x and y are orthogonal , and M 1 N to indicate that M and N are

orthogonal . The orthocomplement of a set A CH is the set { x | ( x , A ) = 0 } .

Page 1779

A set A is called an orthonormal basis for the linear

orthonormal set contained in N and if | = ( x , y ) , a c . VEA 12 THEOREM . Every

closed linear

A set A is called an orthonormal basis for the linear

**manifold**N in H if A is anorthonormal set contained in N and if | = ( x , y ) , a c . VEA 12 THEOREM . Every

closed linear

**manifold**in H contains an orthonormal basis for itself . PROOF .Page 1912

( See also Functional ) Linear

operator , ( 36 ) . ( See also Bspace ) Linear space , 1 . ... 4 ( 121 ) remarks on , (

387 - 388 ) separable

3 , III .

( See also Functional ) Linear

**manifold**, ( 36 ) . ( See also**Manifold**) Linearoperator , ( 36 ) . ( See also Bspace ) Linear space , 1 . ... 4 ( 121 ) remarks on , (

387 - 388 ) separable

**manifolds**in , III . 8 . 5 ( 168 ) , III . 9 . 6 ( 169 ) study of , III .3 , III .

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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additive Akad 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 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 matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero