subspace

**surjective** · March 29th 2011, 03:55 PM

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 $L^{p}(\mathbb{R})$ ?

**Drexel28** · March 29th 2011, 05:34 PM

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.