What is the definition of $C_{c}^\infty (\mathbb{R}^{n})$ spaces?

Question

What is the definition of $C_{c}^\infty (\mathbb{R}^{n})$ spaces? Give Examples

Answer 1

If $X$ is a differentiable manifold, $C^\infty_c(X)$ the set of $C^\infty$ functions $X\to\Bbb R$ with compact ("${}_c$") support. For instance, when $X$ is compact it's just $C^\infty(X)$. When $X=\Bbb R$, a typical example of one of its elements is $$g(x)=\begin{cases}\exp\frac1{x^2-x}&\text{if }x\in(0,1)\\ 0&\text{if }x\in(-\infty,0]\cup[1,\infty)\end{cases}$$

And when $X=\Bbb R^n$, another typical example is $$G(x)=g(x_1)\cdot g(x_2)\cdots g(x_n)$$ where $g$ is the previous function.