Let X be a locally compact Hausdorff space and C <mrow class="MJX-TeXAtom-ORD">

Isabela Sherman

Isabela Sherman

Answered question

2022-05-21

Let X be a locally compact Hausdorff space and C b the set of all continuous functions with support compact and C 0 ( X ) the set of all functions with compact support. The dual of C b is the dual of C 0 ( X )?

Answer & Explanation

tradirasi

tradirasi

Beginner2022-05-22Added 6 answers

Any normed space's dual and its completion's dual are compatible.
Concretely, finishing your spaces doesn't give you access to new bounded linear functionals. A bounded linear functional on a normed space is the cause of this X extends uniquely to the completion X ¯ . Indeed, if f : X  C is linear and bounded, and x n  x, then
| f ( x n )  f ( x m ) | = | f ( x n  x m ) |   f    x n  x m  .
Since { x n } is Cauchy, so is the number sequence { f ( x n ) }. So there exists a limit f ~ ( x ) = lim n f ( x n ). Afterward, one confirms that this cap is distinct, which makes f ~ linear, and it is also easy to check that f ~ is bounded, with  f ~  =  f , and that f ~ = f on X.

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