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Consider the subset S = $L^\infty(\mathbb{R}) \cap L^2(\mathbb{R})$ of $L^\infty(\mathbb{R})$. We know that $L^\infty(\mathbb{R})$ is a Banach space and complete. Is this subset $S$ complete under the $\|\cdot\|_\infty$ distance metric? If yes, where can I find a proof for it?

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No. Since $S$ is a subset of the complete space $L^\infty$, it will be complete if and only if it is closed. But this is not true since any continuous function of compact support is in $S$, and the closure in the $L^\infty$-norm of the continuous functions of compact support is the set $C_0$ of continuous functions vanishing at infinity. It's fairly simple to furnish such a function which is not in $L^2$. For example, $f(x) = \frac{1}{(1 + |x|)^{1/2}}$ will do the trick.

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    Remark: An unclosed subspace is incomplete regardless of whether the containing space is complete.2012-07-01
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If $f_n(x)=\chi_{[-n,n]}(x)\cdot\dfrac{1}{\sqrt{1+|x|}}$, then $(f_n)$ is a Cauchy sequence with no limit.

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    @Thomas: I had initially been thinking of $f$ which vanish at $\infty$, hence the example I gave, then I apparently stopped thinking. Thanks for pointing out the errors, which I will now remove.2012-07-01