I have a curiosity,
For a given function space $V$, and a metric $d(\cdot,\cdot)$ is it possible to construct a non-empty subspace $W$ such that $(W,d)$ is a Banach space?
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http://math.stackexchange.com/questions/1326890/is-a-banach-space-also-a-metric-space – m_gnacik May 17 '16 at 10:44
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You can assume $d$ is induced by a norm $||\cdot||_V$ – user8469759 May 17 '16 at 10:52
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1So $V$ is a normed space already? Then take any finite dimensional $W$ will do. – May 17 '16 at 11:38
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Because every finite dimensional space is isomorphic to $\mathbb{R}^n$? – user8469759 May 17 '16 at 12:00
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@lukkio : Yes, for every finite dimension normed space, the norm is equivalent to the standard euclidean norm, so it is also complete. – May 18 '16 at 21:08