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I am going through theorems that refer to the completion of a metric space. The following theorem states: If $E$ is a metric space there is a complete metric space $\hat E$ and a function $j:E\to \hat E$ that preserves the metric such that $j(E)$ is dense in $\hat E$. After the proving that the sequences converge in both theorems and that the function preserves the metric. After the proof an observation says that $\hat E$ completes $E$. How so? For $\hat E$ to be complete it needs $E$ to be complete right? What is the intuition behind the completion of a metric space such as $\Re[a,b]$, the space of Riemann integrable functions? Thanks for reading!

Umberto P.
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Pedro Gomes
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  • You probably mean to claim that "the sequences converge in both spaces", not "in both theorems". – Mikhail Katz Mar 15 '17 at 16:08
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    To address your first question, for $\hat E$ to be complete it is not necessary that $E$ be complete. For a counterexample, take $E=(0,1)$ which is not complete. It's completion is $\hat E = [0,1]$ which is complete. – Lee Mosher Mar 15 '17 at 16:11

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It is not true that the sequences converge in both spaces $E$ and $\hat E$. For example, the sequence $3, 3.1, 3.14, 3.141, 3.1415, \dots$ does not converge in $\mathbb Q$ but it does converge in its completion $\hat {\mathbb{Q}}=\mathbb{R}$.

Mikhail Katz
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If E is already complete then the completion of E is just E. The completion of E is always complete by definition (it's stated in your theorem).

I always think of completions by analogy to the reals being the completion of the rationals. You're just filling in the gaps (adding all of the limit points).

Completions can be useful (like with Riemann integrable functions), to prove existence and uniqueness theorems. For example you would prove a sequence has a limit by proving that it's a Cauchy sequence. Unless your space is complete, Cauchy sequences may not have a limit. The completion is a way of getting around this issue with non-complete spaces.

Usually it's nice to have a characterization of your completion (like thinking of reals as infinite decimals instead of some kind of quotient of the space of sequences of rationals). For another example, hopefully the completion of the Riemann integrable functions can be characterized as some space of functions.

mathematician
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