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For $p=2$, $(C[0,1],\|\cdot\|_{p})$ is not a complete metric space and its closure is $L^{p}[0,1]$?

I am curious as to whether this is true for all $p<\infty$?

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Yes, at least for $1\le p<\infty$. The references must be legion; here is one such: Proposition 21.1 on page 258 in Emanuele DiBenedetto: Real Analysis. I expect it is also true for $0<p<1$, but those spaces are much harder to study.

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