Explanations
In metric spaces, the completeness is a property of the metric. It is not a property of the topological space itself. If you move on to an equivalent metric (that is a metric which induces the same topology), the completeness can get lost. Regarding two equivalent norms on a normed vector space, however, one of them is complete if and only if the other one is complete. Therefore, in the case of normed vector spaces, the completeness is a property of the norm topology, which does not depend on the specific norm.
Read more about this topic: Banach Space
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