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

Question

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

My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So, $(L^{1},\|\|_{1})$ is vector space. To show that is a Banach space, note that

Prop. 1: Let $(f_{n})_{n\in\mathbb{N}}\subset\mathcal{L}^{1}$ (where $\mathcal{L}^{1}=\mathcal{L}^{1}(X,\tau,\mu)$ is the space of all integrals function), such that $\sum_{i}{\|f_{n}\|}_{1}<\infty$, then the sequences $\left(\sum_{n=1}^{N}{f_{n}}\right)_{n\in\mathbb{N}}$ converges almost everywhere to the integral function, that we called $\sum_{n\in\mathbb{N}}{f_{n}}$. Furthermore $$\sum_{n\in\mathbb{N}}{\int{f_{n}d\mu}}=\int{\sum_{n\in\mathbb{N}}{f_{n}d\mu}}\quad\lim_{N\to\infty}{\|\sum_{n=1}^{N}{f_{n}}-\sum_{n\in\mathbb{N}}{f_{n}}\|_{1}}=0$$

Prop. 2: Let $(E,\|\|)$ a normed vector space. Then $E$ is a Banach space if and only if for all $(e_{k})_{k\in\mathbb{N}}\subset E$ such that $\sum_{k\in\mathbb{N}}{\|e_{k}\|}<\infty$, $\left(\sum_{k=1}^{n}{e_{k}}\right)_{n\in\mathbb{N}}$ converges in $E$

We take the proposition 2. Let $(f_{n})_{n\in\mathbb{N}}\subset L^{1}$ such that $\sum_{n}{\|\hat{f}_{n}\|_{1}}<\infty$.

Let $f_{n}\in\hat{f}_{n}$, then $\|f_{n}\|_{1}=\|\hat{f}_{n}\|_{1}$ and then $\sum_{n}{\|f_{n}\|_{1}}<\infty$, by prop. 1, there exist $F\in\mathcal{L}^{1}$ such that $\|\sum_{n=1}^{N}{f_{n}-F\|_{1}}\to 0$, and then $\|\sum_{n=1}^{N}{\hat{f}_{n}-F\|_{1}}\to 0$. Finally note that $\hat{F}\in L^{1}$ is a Banach space. This was my answer for $L^{1}$ space, but how I prove the general statement for $L^{p}$ (I want to do a similar response).

Edit: I'm stuck in the problem, any idea or hint is appreciated.Thanks!!

Answer 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.

Answer 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 $L^\infty$