Currently I am studying for my exam of Real Analysis, however there is one thing that I do not seem to get. Given:
$$
\mathrm{Supp}(f):=\overline{\{x\in\mathbb{R}^n:f(x)\neq 0\}}
$$
the support of $f$. If the support of $f$ is compact, then we have a compact support. Define:
$$
C_c(\mathbb{R}^n):=\{f:\mathbb{R}^n\to\mathbb{R}: f \text{ continous }, \mathrm{Supp}(f) \text{ compact}\}.
$$
Then $C_c(\mathbb{R}^n)$ is a vectorspace.
For this we need to prove two things:
- $\forall \alpha\in\mathbb{R}\Rightarrow \alpha\cdot f\in C_c(\mathbb{R}^n) \text{ if } f\in C_c(\mathbb{R}^n)$
- if $f,g\in C_c(\mathbb{R}^n)\Rightarrow f+g\in C_c(\mathbb{R}^n)$
However I already get stuck proving the first identity. My problem here is that I do not know if $\mathrm{Supp}(f)$ is compact. Could I get help proving that this is actually a vector space?
Thanks a lot!
Best Answer
The support of $\alpha f$ coincides with the support of $f$ for $\alpha\neq 0$, and the support of $f+g$ is contained in the union of $\mathrm{supp} (f)$ and $\mathrm{supp} (g)$ which shows that $\mathrm{supp} (f+g)$ is bounded. In $\mathbb{R}^n$, any bounded closed set is already compact.