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Let $X$ be normed space and let $B_X$ be the closed unit ball of $X$ is. Then why $X=\{0\}$ if $B_X$ is finite?

Thank you!!

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If $X\neq\{0\}$, take $v\in X\setminus\{0\}$. Then all vectors $\frac t{\|v\|}v$ with $t\in[-1,1]$ belong to the closed unit ball.

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(Assuming the field is something like $\mathbb{R}$)

If $x \in B_X$ is nonzero, then $cx \in B_X$ for any $c \in [0,1]$.

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Assuming that $X$ is considered as a vector space over $\mathbb{R}$, then $X$ must be a finite set itself, otherwise you can always scale the points of $X$ by $\lambda$ that can be chosen for any $v \in X$ such that $|\lambda| \leq \frac{v}{\|v\|}$ and you will find infinite points in the unit ball. Therefore, $X$ is finite and the only finite $\mathbb{R}$-vector space is $\{0\}$. Hence, $X=\{0\}$.

The same thing works for $\mathbb{C}$ too.

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