The meaning of notation $\|x - x^*\|$

Question

I was just wondering what $\|x - x^*\|$ in the following equation means:

$$B(\epsilon) = \{x : \|x - x^*\|<\epsilon\} $$

Thanks.

Answer 1

It represents an open ball. In a normed space, $B(\epsilon)$ represents the set of all $x$ that are $\epsilon$-distant (under the norm $|| \cdot ||$ or generally under some metric) from a fixed point $x^*$.

$B$ saysthat it is a ball. $\epsilon $ denotes the distance. $()$ and $[]$ generally stand for open and closed balls which contain $\lt $ and $\le $ respectively in the definition. You can make out the difference.

Bartle's Elements of Real Analysis has a nice explanation on ballsin Euclidean space which this seems to refer to.