Measure Theory/L2 Is Complete

L² Is Complete

Our goal in this section, let us repeat, is to prove the completeness of $L^2(E)$. Let us now establish the language needed to express this formally.

From now on we simply say "converges" instead of "converges in distance" since there will be no possibility of confusion; and we say "Cauchy" instead of "Cauchy in distance".

Convergent Subsequences

The following two exercises would be good mere warm-ups for the concept of convergence. However, more than that, we will use them in the proof of completeness.

Let $⟨f_n⟩$ be a sequence of functions in $L^2(E)$ which converges to $f\in L^2(E)$.

Show that $⟨f_n⟩$ is Cauchy.

The proof should be familiar from introductory analysis, in particular the proof that if a sequence of real numbers is convergent then it is Cauchy.

Let $⟨f_n⟩$ be a sequence of functions in $L^2(E)$ which is Cauchy and has some subsequence $⟨f_{n_k}⟩$ which converges to $f\in L^2(E)$.

Prove that $⟨f_n⟩$ converges to f.

Absolutely Convergent Series

In this section we will study a fact which actually holds for any arbitrary normed space, and then apply it to $L^2(E)$, to help us infer completeness. Because we situate the proof in a more abstract space, I find that this actually helps to make the proof easier to understand since it blots out certain details which obscure the logic if we tried a more direct proof.

Of course we will start from a sequence of vectors (functions $f_n\in L^2(E)$) and we would like to show that there is a vector (function $f\in L^2(E)$) such that $⟨f_n⟩$ converges to f.

So what does this have to do with series? Nothing, intrinsically. But one can reformulate the series by

$f_{n+1}=f_1+\sum _{k=1}^n(f_{k+1}-f_k)$

which essentially says that one may "add in" $f_1$ and then subtract it out and add in $f_2$ and so on up to $f_{n+1}$. Since we are hoping that the sequence converges then it should hopefully be the same if we show that the series converges.

But as such, if we can think clearly about sums then perhaps it can tell us about our sequence.

Prove that, for any normed vector space V, it is complete if and only if every absolutely summable sequence is summable.

(Note that the condition "every absolutely summable sequence is summable" is a lot like the "series version" of the idea that every Cauchy sequence is a convergent sequence. Both Cauchy sequences and absolutely summable series do not specify any vector to which the sequence converges. But in a complete space, we are guaranteed that there must exist some vector to which it converges.)

Hint: Going from completeness to the equivalent property isn't much more than unpacking definitions and using the triangle inequality.

The relatively harder part is the converse, which of course is the direct that we will actually need for our intended application.

To help with this direction I will show you a failed attempt to prove this direction, and your job will be to fix it into a correct proof.

Failed attempt: Suppose that V is a vector space with the property that every absolutely summable series is summable. Let ${\overrightarrow{v}}_n\in V$ be a Cauchy sequence of vectors. Clearly we need to find some absolutely summable series in order to exploit its implied summability.

We try to consider, for each natural number $k\in \mathbb{N}$ the natural number $N_k\in \mathbb{N}$ for which, for all $N_k\le m,n$ we have

$\Vert {\overrightarrow{v}}_m-{\overrightarrow{v}}_n\Vert <\frac{1}{k}$

In particular we have a subsequence ${\overrightarrow{V}}_{N_k}$ determined by the above, such that

$\Vert {\overrightarrow{v}}_{N_{k+1}}-{\overrightarrow{v}}_{N_k}\Vert <\frac{1}{k}$

1. Try (and fail) to show that the series $\sum _{k=1}^{\mathrm{\infty }}({\overrightarrow{v}}_{N_{k+1}}-{\overrightarrow{v}}_{N_k})$ is absolutely summable.

2. Make a correction of the above to obtain an absolutely summable series.

3. Argue that the subsequence (not series) which you obtain in (2.) converges to some vector.

4. Use the result of Exercise 2.

Use Exercise 3. to prove the completeness of $L^2(E)$.

Here are some guiding steps.

Let $f_n\in L^2(E)$ be any sequence of functions which is absolutely summable, i.e.

$\sum \Vert f_n{\Vert }_2=M<\mathrm{\infty }$

We need to find a function to which $\sum f_n$ converges. The natural guess is the pointwise limit of the partial sums.

$F(x)=\sum f_n(x)$

But how do we show that ${\Vert F-\sum f_n\Vert }_2\to 0$, and that $F\in L^2(E)$.

1. Well first thing's first: Argue that F is measurable.

2. Argue that $F\le \sum |f_n|$ (each understood as pointwise limits).  In order to help with notation, let $g(x)=\sum |f_n(x)|$ so that you are proving $F(x)\le g(x)$.  

(Note that, although this seems like a relatively natural thing to do, this is currently more like a "brainstorming" idea. As of right now we do not know that g exists a.e. We will prove that below.)

The point of using g is the hope that it may be easier to prove that $g\in L^2(E)$ and then use this to show that $F\in L^2(E)$. This is a common dance that we do in situations like this.

But then how do we show that $\underset{E}{\int }{g}^2<\mathrm{\infty }$? It must in some way relate to the assumption $\sum \Vert f_n{\Vert }_2<\mathrm{\infty }$, and it is natural to try to obtain this through some sort of limiting process. Well this is precisely what our convergence theorems are good for!

3. Define $g_n=\sum _{k=1}^n|f_k|$ pointwise.  Use the triangle inequality and the absolute summability of $⟨f_n⟩$ to prove that 
  $\Vert g_n{\Vert }_2\le M$

4. Apply Fatou's to show that 
  $\underset{E}{\int }{g}^2\le M^2$

5. Use the above to show that $F\in L^2(E)$ and moreover 
  ${|\sum _{k=1}^n{f}_k(x)-F(x)|}^2\le 4(g(x){)}^2$

6. Infer that ${|\sum _{k=1}^n{f}_k(x)-F(x)|}^2\to 0$ as $n\to \mathrm{\infty }$. a.e., and draw the desired conclusions from this.