# Continuous functions spaces

Recently I had to dive into abstract mathematics to understand deeply finite element method (I am an engineer not a mathematician). In some examples of linear spaces it appeared the space:

$C_{0}^{\infty}(\Omega)$ with $\Omega\subset$ in $\mathbb{R}^{d}$

The context says that this subspace of $\mathbb{R}^{d}$ is dense in $L^{p}(\Omega)$

Is this the set of continuous functions and derivatives that converges to '0'?

• $C^{\infty}$ is the set of infinitely differentiable functions
– Alex
Commented Mar 4, 2015 at 11:34
• I would guess this refers to the smooth functions that vanish at infinity, although it is always possible this particular author uses $0$ to denote some other property. Commented Mar 4, 2015 at 11:46
• And where did that space appear? It's very probably defined somewhere in what you're reading... Commented Mar 5, 2015 at 8:57

Depending on whom you ask, the notation $C_0^\infty$ means one of two things:
1. The set of all infinitely differentiable functions $f$ such that $f=0$ outside of some compact set $K$
2. The set of all infinitely differentiable functions $f$ such that $f$, and every derivative of $f$, tend to $0$ at infinity (or on the boundary of the domain, if we consider a domain instead of all $\mathbb{R}^n$).
People who subscribe to interpretation #2 use $C_c^\infty$ for the space from #1.
• It could also be the space of infinitely differentiable functions $f$ satisfying $f(0)=0$. Commented Mar 4, 2015 at 14:25