Continuous functions spaces

Question

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'?

Answer 1

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.

The space #2 is strictly larger than the space #1. One would have to see the book/paper to infer from the context which one is meant.