subspace

• March 29th 2011, 04:55 PM
surjective
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 $L^{p}(\mathbb{R})$?
• March 29th 2011, 06:34 PM
Drexel28
Quote:

Originally Posted by surjective
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.
• March 30th 2011, 05:57 AM
surjective
Hey,

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

Thanks
• March 30th 2011, 08:36 AM
Opalg
Quote:

Originally Posted by surjective
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.