How will we prove
that the closed unit ball in $\ell^2$ is closed, bounded, convex but not compact
[Math] Closed Unit ball in $\ell^2$ is not compact
compactnessfunctional-analysislp-spacesreal-analysis
compactnessfunctional-analysislp-spacesreal-analysis
How will we prove
that the closed unit ball in $\ell^2$ is closed, bounded, convex but not compact
Best Answer
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).