Show that $(L^{p},\|\|_{p})$ is a Banach space.

Solution 1:

Let's break things down into steps:

1. $L^p$ is a vector space. (Follows from their definition.)

2. They are normed vector spaces: The $L^p$ norm, by definition, is a finite, nonnegative real number for given $f \in L^p$.

   2.1 $\|f\|_p=0$ iff $f=0$ in $L^p$. This follows from the fact that if $f\neq 0$ on a set of positive measure, then $\int |f|^p >0.$

   2.2 The triangle inequality: For $p=2$ this follows from Cauchy-Schwarz. For general $p$ we use Hölder inequality, which is Cauchy-Schwarz with $1/p$ and $1/q$ replacing $2$.

3. So the last part that needs to be proved is completeness with respect to the above norm. I found a PDF by googling the other day, which had a complete proof. It uses monotone convergence once and dominated convergence once. But the proof runs smoothly, no real trick. It started with a Cauchy (in this norm of course) sequence of functions, and then extracts a convergent subsequence (again in $L^p$ norm to a limit WHICH IS in $L^p$, i.e. Any Cauchy sequence converges.

Solution 2:

Recall that any series $\sum_{n=1}^\infty a_n$ of scalars is convergent if it is absolutely convergent (i.e. if $\sum_{n=1}^\infty |a_n| < \infty$). This fact turns out to be closely related to the fact that the field of scalars ${\Bbb C}$ is complete. This can be seen from the following result:

(This is essentially your Proposition 2)

Let $(V, \| \|)$ be a normed vector space (and hence also a metric space and a topological space). Show that the following are equivalent:

• $V$ is a complete metric space (i.e. every Cauchy sequence converges).

• Every sequence $f_n \in V$ which is absolutely convergent (i.e. $\sum_{n=1}^\infty \|f_n\| < \infty$), is also conditionally convergent (i.e. $\sum_{n=1}^N f_n$ converges to a limit as $N \to \infty$).

Once the above result is established, we can do the proof as follows. It suffices to show that any series $\sum_{n=1}^\infty f_n$ of functions in $L^p$ which is absolutely convergent, is also conditionally convergent. In the case $1 \leq p < \infty$, we write $M := \sum_{n=1}^\infty \|f_n\|_{L^p}$, which is a finite quantity by hypothesis. By the triangle inequality, we have $$ \| \sum_{n=1}^N |f_n| \|_{L^p} \leq M $$ for all $N$. By monotone convergence, we conclude $$ \| \sum_{n=1}^\infty |f_n| \|_{L^p} \leq M. $$ In particular, $\sum_{n=1}^\infty f_n(x)$ is absolutely convergent for almost every $x$. Write the limit of this series as $F(x)$. By dominated convergence, we see that $\sum_{n=1}^N f_n(x)$ converges in $L^p$ norm to $F$, and we are done.

For the case $p=\infty$, you can see these two questions:
In a proof of the completeness of $L^\infty$
In Rudin's proof of the completeness of $L^\infty$