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Suppose that X is a Banach space. Let $X_0$ be a dense subspace of X. Assume that $X_0$, when normed by the norm it inherits from X, is also a Banach space. Prove that $X_0 =X$.
For this problem, do I only need show that X is contained in $X_0$? Thanks.

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  • $\begingroup$ That's right. But it might not be the easiest starting point. A more amenable one might be to show $X_0$ is closed (why?) $\endgroup$ – Daron Mar 24 '17 at 23:04
  • $\begingroup$ I think it is unnecessary to show $X_0$ is closed. $\endgroup$ – Dean Young Mar 24 '17 at 23:32
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Yes, showing $X \subset X_0$ would be sufficient, since $X_0 \subset X$.

Let's phrase the result in a greater generality:

Lemma: A complete dense subset $X_0$ of a metric space $X$ must be $X$ itself.

Each element $x$ of $X$ not in $X_0$ is a limit point of $X_0$ since $X_0$ is dense in $X$. Thus $x$ can be written as the limit of a Cauchy sequence in $X_0$. As $X_0$ is complete it must contain $x$.

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