I am solving tutorials of an institution which I am not a student as our faculty doesn't gives any. Please give me some hint in solution of this problem. Problem is - let X be a normed linear space. Prove that X is a Banach space iff $\,\{ x: \|x\| = 1 \}\,$ is complete.
2026-03-29 18:30:25.1774809025
Regarding proving a result in Banach space
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Sketch of proof.
If $\{x_n\}\subset X$ is Cauchy, and DOES NOT converge to zero, then $\{\|x_n\|\}\subset \mathbb R$ is also Cauchy, and hence convergent, say to $a>0$.
Then, it is not hard to show that $\{\|x_n\|^{-1}x_n\}\subset\{x\in X: \|x\|=1\}$ is also Cauchy, and hence convergent. Say $\|x_n\|^{-1}x_n\to y$.
It is now easy to show that $x_n\to ay$.