Is it the same to say that a random vector space is a Banach space, and to say that a normed vector space is complete?

If so, when asked to prove that some space with specific norm is complete, is it enough to show that every Cauchy Sequence {$ x_n $} is convergent?

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    $\begingroup$ A Banach space is a normed vector space which is complete with respect to the metric induced by the norm. A complete metric space is a metric space in which every Cauchy sequence converges. $\endgroup$ – Ashwin Iyengar Oct 6 '16 at 0:26
  • $\begingroup$ So a complete metric space IS a Banach space with respect to the metric. Right? $\endgroup$ – cryptow Oct 6 '16 at 16:52
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    $\begingroup$ No: a metric space, in general, doesn't have to be a vector space, and a Banach space (a complete normed vector space) is always a vector space. A Banach space is defined as a pair $(V, ||\cdot||)$ where $V$ is a vector space and $||\cdot||$ is a norm on that vector space, such that the metric space $(V,d(\cdot,\cdot))$ is complete, where $d(x,y) := ||x-y||$. $\endgroup$ – Ashwin Iyengar Oct 6 '16 at 18:03

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