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Thread: subspace

  1. #1
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    subspace

    Hey,

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

    Can I use the usual three conditions to show that $\displaystyle C_{c}(\mathbb{R})$ is a subspace. But how would that show that $\displaystyle C_{c}(\mathbb{R})$ is a subspace of specifically $\displaystyle L^{p}(\mathbb{R})$?
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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 $\displaystyle C_{c}(\mathbb{R})$ is a subspace of $\displaystyle L^{p}(\mathbb{R})$.

    Can I use the usual three conditions to show that $\displaystyle C_{c}(\mathbb{R})$ is a subspace. But how would that show that $\displaystyle C_{c}(\mathbb{R})$ is a subspace of specifically $\displaystyle L^{p}(\mathbb{R})$?
    Is $\displaystyle C_c\left(\mathbb{R}\right)$ 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 $\displaystyle \text{supp}(cf)=\text{supp}(f)$ and $\displaystyle \text{supp}(f+g)\subseteq \text{supp}(f)\cup\text{supp}(g)$ so that $\displaystyle \overline{\text{supp}(f+g)}$ is a closed subspace of $\displaystyle \overline{\text{supp}(f)}\cup\overline{\text{supp} (g)}$ and since this superset is compact it follows that $\displaystyle \overline{\text{supp}(f+g)}$ is compact.
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  3. #3
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    Hey,

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

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