What does $\mathcal{C}^{\infty}_{\mathcal{C}}(Z)$ mean?

Question

In my course material I have the following notation $$ \varphi \in \mathcal{C}^{\infty}_{\mathcal{C}}(Z) $$ where $Z⊂R^{n}$ is a bounded open set. I was wondering what does this notation mean? What is the set $\varphi$ belongs into? Please, help me :)

Is it really written with calligraphic C at both places? – md2perpe Jan 06 '21 at 23:07 — md2perpe, Jan 06 '21 at 23:07

score 3 · Answer 1 · answered Jan 06 '21 at 21:11

3

$\mathcal{C}^{\infty}_{\mathcal{C}}(Z)$ is the set of infinitely differentiable functions defined on $Z$ with compact support, i.e. they are zero outside some compact set.

answered Jan 06 '21 at 21:11

DMcMor

9,407

1

Thank you very much! – machniik Jan 06 '21 at 21:15
I've never seen it with a calligraphic C in either place. It's normally written $C_c^\infty(Z)$. But styles can vary. – md2perpe Jan 06 '21 at 22:23
@md2perpe I hadn't either, I'm only following the convention used by machniik. – DMcMor Jan 06 '21 at 22:25

What does $\mathcal{C}^{\infty}_{\mathcal{C}}(Z)$ mean?

1 Answers1