## Linear Operators: General theory |

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Page 54

Since the

proof is complete . Q . E . D . We now ... A linear

another is continuous if and only if it

Proof .

Since the

**mapping**x + x / k , is a homeomorphism of X with itself , by 10 ( ii ) , theproof is complete . Q . E . D . We now ... A linear

**mapping**of one F - space intoanother is continuous if and only if it

**maps**bounded sets into bounded sets .Proof .

Page 493

Conversely , if ( a ) and ( b ) are satisfied for the

• ) * * , then it follows that for each fixed f e C ( S ) the

) x * is continuous in the X topology of X * and therefore ( V . 3 . 9 ) is generated ...

Conversely , if ( a ) and ( b ) are satisfied for the

**mapping**which sends q * into u (• ) * * , then it follows that for each fixed f e C ( S ) the

**mapping*** * → [ st ( s ) u ( ds) x * is continuous in the X topology of X * and therefore ( V . 3 . 9 ) is generated ...

Page 513

12 If H is a Hilbert space , the

with either the uniform or weak operator topology . By considering the sequence {

An } defined in Exercise 11 , show that this

12 If H is a Hilbert space , the

**mapping**T → T * of B ( H ) into itself is continuouswith either the uniform or weak operator topology . By considering the sequence {

An } defined in Exercise 11 , show that this

**mapping**is not continuous in the ...### What people are saying - Write a review

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### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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