Open Ball Centred at Origin in Normed Vector Space is Symmetric

Theorem

Let $\struct {X, \norm \cdot}$ be a normed vector space.

Let $\map B {0, r}$ be the open ball in $X$ centered at $0$ with radius $r$.

Then $\map B {0, r}$ is symmetric.

Proof

Let $x \in \map B {0, r}$.

Then:

$\norm x < r$

We then have:

$\norm {-x} = \cmod {-1} \norm x = \norm x < r$

So $-x \in \map B {0, r}$.

So $\map B {0, r}$ is symmetric.

$\blacksquare$