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How will we prove that the closed unit ball in $\ell^2$ is closed, bounded, convex but not compact

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    [Proving that the unit ball in $\ell^2(\mathbb{N})$ is non-compact](http://math.stackexchange.com/q/115344)2012-11-26

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In addition to the above, convexity is straightfoward. Pick $x,y$ in the closed unit ball and $0 < t <1$. We need to show that $||tx + (1-t)y|| \le 1$. This follows from the observation:

$||tx + (1-t)y|| \le ||tx|| + ||(1-t)y|| = t||x|| + (1-t)||y|| \le max(||x||,||y||) \le 1 $

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Denote by $e_n \in \ell^2$ the sequence $(0, \ldots, 0, 1, 0, \ldots)$ with the $1$ at position $n$. Then $$\|e_n - e_m\|_2 = \sqrt 2, \qquad n \ne m $$ Hence $(e_n)$ cannot have any subsequence which is Cauchy. (The closed unit ball is moreover bounded by 1 and closed).

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    how $\sqrt 2 $ has come ??? i think it should be $$\|e_n - e_m\|_2 = 1$$....@ martini can u elaborate more...?2018-04-12