For $f:\mathbb R^N\to\mathbb C$, we write $f\in C^\infty(\mathbb R^N,\mathbb C)$ if $f$ is infinitely partially differentiable.
Using multi-index notation, can I transrate it to
$f\in C^\infty(\mathbb R^n,\mathbb C)$ if for all multi-index $\alpha$ and $x\in\mathbb R^N$, $\partial^\alpha f(x)$ exists
?
I'm not sure because the books I read doesn't write this expression.