Open Ball in Normed Vector Space is Connected

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Theorem

Let $V$ be a normed vector space with norm $\norm {\,\cdot\,}$ over $\R$ or $\C$.

An open ball in the metric induced by $\norm {\,\cdot\,}$ is connected.

Proof

Follows from:

• Open Ball in Normed Vector Space is Path-Connected
• Path-Connected Space is Connected

$\blacksquare$