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A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.
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Explicit description of the closure of a given set
Let $C$ be the subset of $C_b(\mathbb{R})$ given by
$$C:=\{f\in C_b(\mathbb{R}):\ \exists f'\in C_b(\mathbb{R})\}$$
Now I want to take the closure of this set with respect to the supremum norm on $C_ …