Let $X$ be a non-trivial (other than singleton $x$) normed space. Prove that $X$ is a Banach space if and only if $\{x \in X \mid \|x\| = 1 \}$ is complete.
2026-03-28 02:04:33.1774663473
A normed space is Banach iff its unit sphere is complete
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Hint
Consider a Cauchy sequence $\{x_n\}$. See what happens with $$\left\{\frac1{\|x_n\|}x_n\right\}$$
(Note that you also have to consider sequences with zeros).