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Math Help - subspace

  1. #1
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    subspace

    Hey,

    I wan't to show that C_{c}(\mathbb{R}) is a subspace of L^{p}(\mathbb{R}).

    Can I use the usual three conditions to show that C_{c}(\mathbb{R}) is a subspace. But how would that show that C_{c}(\mathbb{R}) is a subspace of specifically L^{p}(\mathbb{R})?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by surjective View Post
    Hey,

    I wan't to show that C_{c}(\mathbb{R}) is a subspace of L^{p}(\mathbb{R}).

    Can I use the usual three conditions to show that C_{c}(\mathbb{R}) is a subspace. But how would that show that C_{c}(\mathbb{R}) is a subspace of specifically L^{p}(\mathbb{R})?
    Is C_c\left(\mathbb{R}\right) continuous functions with compact support? If so, then what particularly are you having trouble with, you know that the sum of two continuous functions is continuous as is the product of a continuous function by a scalar, thus it suffices to prove that the same is true for functions with compact support. But, it's clear that \text{supp}(cf)=\text{supp}(f) and \text{supp}(f+g)\subseteq \text{supp}(f)\cup\text{supp}(g) so that \overline{\text{supp}(f+g)} is a closed subspace of \overline{\text{supp}(f)}\cup\overline{\text{supp}  (g)} and since this superset is compact it follows that \overline{\text{supp}(f+g)} is compact.
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  3. #3
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    Hey,

    And C_{c}(\mathbb{R}) is a subset of Lp(\mathbb{R})? Then comes the explanation you gave above. Right?

    Thanks
    Last edited by surjective; March 30th 2011 at 05:47 AM.
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  4. #4
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    Quote Originally Posted by surjective View Post
    And C_{c}(\mathbb{R}) is a subset of L_p(\mathbb{R})?
    To check that, you need to verify that if f\in C_{c}(\mathbb{R}) then |f|^p is integrable.
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