Vector space notation

Question

Explain which of the following vector spaces are subspaces of eachother

${\cal C}(\mathbb{R})$

${\cal C}^n(\mathbb{R})$ (for different $n$)

${\cal C}^{\infty}(\mathbb{R})$.

I do not understand what ${\cal C}$, $n$ and $\mathbb{R}$ means in this context? Could someone clarify?

Answer 1

$\mathcal{C}^n(\mathbb{R})$ usually denotes the set of $n$-times continuously differentiable functions from $\mathbb{R}$ to $\mathbb{R}$ or to $\mathbb{C}$ but that does not really matter.

Answer 2

$C^n(\mathbb R)$ undoubtedly stands for n times continuously differentiable functions defined on the field of real numbers...

$C^{\infty}(\mathbb R)\subset C^m(\mathbb R)\subset C^n(\mathbb R)\subset C(\mathbb R)$ for $m\gt n$ and each $C^n(\mathbb R)$ is easily seen to be a vector space over $\mathbb R$.