= Completion of a normed space
{title2=$X\hookrightarrow\widehat X$}
The completion of a <normed vector space> $X$ is a <Banach space> containing an isometric dense copy of $X$. Construct it from <Cauchy sequences> modulo <sequences> whose difference tends to zero; define the <norm> by $\|[(x_n)]\|=\lim_n\|x_n\|$. Equivalently, the canonical evaluation map into the <bidual space> is an isometry by the <Hahn-Banach theorem>, and the closure of its image supplies a completion.
A <Banach space> is precisely a <normed vector space> in which every <Cauchy sequence> converges in <norm> to an element of the same space. <Completion of a normed space> fills missing limits without changing distances between existing elements. For example, $C([0,1])$ with the <L2 norm> completes to <L2 space> on the interval with <Lebesgue measure>. The bounded <integral> functional extends uniquely to that <completion of a normed space>: define $F(f)=\lim_kF(f_k)$ for any approximating <sequence> of <continuous functions>. Its bound makes this independent of the approximation. The unbounded <point evaluation> has no continuous extension agreeing with its values on all <continuous functions>.https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/3d4cc88026d44a01f936cd6a0aa995cb_MIT18_102s20_lec_FA.pdf[Richard Melrose, Functional Analysis, chapter 1, sections 5–6: completion and continuous extension]{ref}
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