Closed Ball is Path-Connected
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Theorem
Let $V$ be a normed vector space with norm $\norm {\,\cdot\,}$ over $\R$ or $\C$.
A closed ball in the metric induced by $\norm {\,\cdot\,}$ is path-connected.
Proof
Follows from:
- Closed Ball is Convex Set
- Normed Vector Space is Hausdorff Topological Vector Space
- Convex Set is Path-Connected
$\blacksquare$