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.

Definition: converge in distance

Let $f_{n}\in L^{2}(E)$ be a sequence of functions, and $f\in L^{2}(E)$ also. We will say that $\langle f_{n}\rangle$ converges in distance to f if the sequence of distances goes to zero.

$d(f_{n},f)\to 0$ as $n\to \infty$

We say that $\langle f_{n}\rangle$ converges in distance if there exists some function to which it converges in distance.

We will say that the sequence is Cauchy in distance if the sequence of distances is a Cauchy sequence of real numbers. Formally we express this as:

For all $\varepsilon \in {\mathbb {R}}^{+}$ there is a natural number $N\in {\mathbb {N}}$ such that for all $N\leq m,n$ we have

$d(f_{m},f_{n})<\varepsilon$

The following is only a slightly less formal way of saying the same thing:

$d(f_{m},f_{n})\to 0$ as $m,n\to \infty$

We say that a (topological) space is complete (also frequently called "Banach") if every Cauchy sequence converges. (Therefore, for us, we say that $L^{2}(E)$ is complete if every sequence of functions that is Cauchy in distance, converges in distance.)

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.

Exercise 1. If Convergent then Cauchy Let $\langle f_{n}\rangle$ be a sequence of functions in $L^{2}(E)$ which converges to $f\in L^{2}(E)$. Show that $\langle f_{n}\rangle$ 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.

Exercise 2. If Cauchy and Subsequentially Convergent then Convergent Let $\langle f_{n}\rangle$ be a sequence of functions in $L^{2}(E)$ which is Cauchy and has some subsequence $\langle f_{n_{k}}\rangle$ which converges to $f\in L^{2}(E)$. Prove that $\langle f_{n}\rangle$ 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.

Definition: series convergence, absolutely convergent

Let V be a vector space with norm $\|\cdot \|$. Let ${\vec {v}}_{n}\in V$ be a sequence of vectors and also ${\vec {v}},{\vec {w}}\in V$ two vectors. Let $d$ be the distance induced by $\|\cdot \|$.

$d({\vec {v}},{\vec {w}})=\|{\vec {v}}-{\vec {w}}\|$

We say that the series is summable and we write

$\sum _{n=1}^{\infty }{\vec {v}}_{n}={\vec {v}}$

if the sequence of partial sums converges in distance to ${\vec {v}}$. Formally, that is:

$\left\|\sum _{n=1}^{m}{\vec {v}}_{n}-{\vec {v}}\right\|\to 0$ as $m\to \infty$

We say that the series $\sum _{n=1}^{\infty }{\vec {v}}$ is absolutely summable if the series of norms converges:

$\sum _{n=1}^{\infty }\|{\vec {v}}_{n}\|<\infty$

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 $\langle f_{n}\rangle$ 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.

Exercise 3. Complete Iff Absolutely Summable 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 ${\vec {v}}_{n}\in V$ 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 $N_{k}\leq m,n$ we have $\|{\vec {v}}_{m}-{\vec {v}}_{n}\|<{\frac {1}{k}}$ In particular we have a subsequence ${\vec {V}}_{N_{k}}$ determined by the above, such that $\|{\vec {v}}_{N_{k+1}}-{\vec {v}}_{N_{k}}\|<{\frac {1}{k}}$ 1. Try (and fail) to show that the series $\sum _{k=1}^{\infty }({\vec {v}}_{N_{k+1}}-{\vec {v}}_{N_{k}})$ 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.

Exercise 4. L2 Completeness 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. $\sum \|f_{n}\|_{2}=M<\infty$ 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 $\left\|F-\sum f_{n}\right\|_{2}\to 0$, and that $F\in L^{2}(E)$. 1. Well first thing's first: Argue that F is measurable. 2. Argue that $F\leq \sum |f_{n}|$ (each understood as pointwise limits). In order to help with notation, let $g(x)=\sum |f_{n}(x)|$ so that you are proving $F(x)\leq g(x)$. (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 $\int _{E}g^{2}<\infty$? It must in some way relate to the assumption $\sum \|f_{n}\|_{2}<\infty$, 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 $\langle f_{n}\rangle$ to prove that $\|g_{n}\|_{2}\leq M$ 4. Apply Fatou's to show that $\int _{E}g^{2}\leq M^{2}$ 5. Use the above to show that $F\in L^{2}(E)$ and moreover $\left|\sum _{k=1}^{n}f_{k}(x)-F(x)\right|^{2}\leq 4(g(x))^{2}$ 6. Infer that $\left|\sum _{k=1}^{n}f_{k}(x)-F(x)\right|^{2}\to 0$ as $n\to \infty$. a.e., and draw the desired conclusions from this.