## Linear Operators: General theory |

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Page 240

The space bs is the linear space of all sequences x = { an } of

the norm [ x ] = sup / Žail is finite . 11 . The space ... SES A

E - measurable if f - 1 ( A ) €£ for every Borel set A in the range of f . It is clear that

...

The space bs is the linear space of all sequences x = { an } of

**scalars**for whichthe norm [ x ] = sup / Žail is finite . 11 . The space ... SES A

**scalar**function f on S isE - measurable if f - 1 ( A ) €£ for every Borel set A in the range of f . It is clear that

...

Page 256

... the

always given by ( iii ) . To summarize , we state the following definition . 17

DEFINITION . For each i = 1 , . . . , n , let H ; be a Hilbert space with

products ( · , ) i .

... the

**scalar**product in Xi . Thus the norm in a direct sum of Hilbert spaces isalways given by ( iii ) . To summarize , we state the following definition . 17

DEFINITION . For each i = 1 , . . . , n , let H ; be a Hilbert space with

**scalar**products ( · , ) i .

Page 323

A

sequence { In } of simple functions such that ( i ) in ( s ) converges to f ( s ) u -

almost everywhere ; ( ii ) the sequence { Seln ( s ) u ( ds ) } converges in the norm

of X ...

A

**scalar**valued measurable function f is said to be integrable if there exists asequence { In } of simple functions such that ( i ) in ( s ) converges to f ( s ) u -

almost everywhere ; ( ii ) the sequence { Seln ( s ) u ( ds ) } converges in the norm

of X ...

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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