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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?

Calmat
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    By "normed space" you mean over a complete archimedean field, i.e. $\mathbb R$ or $\mathbb C$? Because over ultrametric fields this is certainly wrong. The answers seem to assume reals or complex numbers as base field. – Torsten Schoeneberg Apr 11 '20 at 07:02
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    Yes, in $\mathbb{R}$. – Calmat Apr 11 '20 at 07:08

3 Answers3

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

Masacroso
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Jens Schwaiger
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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.

Masacroso
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Luca Goldoni Ph.D.
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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.

Masacroso
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peek-a-boo
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