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Consider the normed spaces (over the field of real numbers) $X=(\ell_\infty,\|\cdot\|_\infty)$ and $Y=(\ell_\infty,\|\cdot\|)$ where $$\|x\|=\sup_{n\in\mathbf{N}}\frac{|x_n|}{2^n}.$$

How can I show that the closed unit ball in $X$ is compact in $Y$?

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    **Hint:** $Y$ is also known as the Hilbert cube. Show that the topology induced by $\Vert \cdot\Vert$ on $[-1,1]^{\mathbb N}$ is the same as the product topology on this set. Then use Tychonoff.2012-08-31
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    @Sam Is the whole $Y$ a Hilbert cube? Or just that closed ball?2012-09-01
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    @Godot: You are right. It is just the closed ball, of course. Thanks for the correction. =)2012-09-01
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    Hum. How can the unit ball be precompact in a infinite-dimensional normed space? I guess that $Y$ is just a metric space, not a normed space.2012-09-07

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