Linear Operators, Part 1 |
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Page 245
Every linear operator on a finite dimensional normed linear space is continuous .
Proof . Let { b1 , ... , bn } be a Hamel basis for the finite dimensional normed linear
space X so that every x in X has a unique representation in the form x = abit ...
Every linear operator on a finite dimensional normed linear space is continuous .
Proof . Let { b1 , ... , bn } be a Hamel basis for the finite dimensional normed linear
space X so that every x in X has a unique representation in the form x = abit ...
Page 246
Then the dimension of X ** is finite , and , since X is equivalent to a subspace of X
** ( II.3.19 ) , the dimension of X is finite . Hence , from the first part of this proof , X
and X * have the same dimension . Q.E.D. In the preceding lemma it is ...
Then the dimension of X ** is finite , and , since X is equivalent to a subspace of X
** ( II.3.19 ) , the dimension of X is finite . Hence , from the first part of this proof , X
and X * have the same dimension . Q.E.D. In the preceding lemma it is ...
Page 290
It is obvious that \ w * ) = [ gc , and so fæ * 1 = ' glcNow suppose that ( S , E , u ) is
o - finite , and let En be an increasing sequence of measurable sets of finite
measure whose union is S. Using the theorem for L ( En ) = L ( En , E ( En ) , u ) ,
we ...
It is obvious that \ w * ) = [ gc , and so fæ * 1 = ' glcNow suppose that ( S , E , u ) is
o - finite , and let En be an increasing sequence of measurable sets of finite
measure whose union is S. Using the theorem for L ( En ) = L ( En , E ( En ) , u ) ,
we ...
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Contents
Preliminary Concepts | 1 |
The VitaliHahnSaks Theorem and Spaces of Measures | 7 |
B Topological Preliminaries | 10 |
Copyright | |
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