Milnor's proof of Sard's theorem

Question

What does the notation mean in the last step of milnor's proof for Sard's theorem

$f(x+h) = f(x) + R(x+h)$

where $||R(x+h)|| \leq c||h||^{k+1}$

What do the lines || mean here?

Answer 1

If ${\bf x} \in \mathbb{R}^d$ is vector in $\mathbb{R}^d$, we usually use $\|{\bf x}\|$ to represent the Euclidean norm (or "length") of ${\bf x}$. Mathematically, suppose ${\bf x} = (x_1,x_2,\ldots,x_d)$, then $\|{\bf x}\| = \sqrt{\sum_{k=1}^d x^2_k}$