Is the open ball connected in normed space?

Question

I know this is not true for metric spaces. Any help for normed space. If I have a normed space $X$, is any ball around a point $x ∈ X$ so connected?

Answer 1

Yes, it is even convex (given any two points in the ball, the straight line segment joining the points is also contained in the ball) and hence path connected and hence connected.

Answer 2

Every convex set is path-connected and thus connected. Every ball is convex.

Answer 3

It is enough to prove that the unit ball around $0$ is convex. Let be $a,b$ two points of this ball such that $$ \left\| a \right\| \leqslant \left\| b \right\| \leqslant 1 $$ If $0 \leq t \leq 1$ we have that $$ \left\| {at + \left( {1 - t} \right)b} \right\| \leqslant \left\| {at} \right\| + \left\| {\left( {1 - t} \right)b} \right\| = t\left\| a \right\| + \left( {1 - t} \right)\left\| b \right\| \leqslant t\left\| b \right\| + \left( {1 - t} \right)\left\| b \right\| = \left\| b \right\| \leqslant 1 $$ thus each point of the segment $[a,b]$ belongs to the unit ball.