The following is from Folland's Real Analysis, Theorem 6.6. The theorem states:
$\underline{\text{Theorem } 6.6:}$ For $1 \leq p < \infty$, $L^{p}$ is a Banach space.
The proof makes use of Theorem 5.1 which is as follows:
$\underline{\text{Theorem } 5.1:}$ A normed vector space $X$ is complete iff every absolutely convergent series in $X$ converges.
$\underline{\text{The Proof of Theorem 6.6}:}$
We use Theorem 5.1. Suppose $\left \{ f_{k} \right \} \subset L^{p}$ and $\displaystyle \sum_{k=1}^{\infty} ||f_{k}||_{p} = B < \infty$. Let $\displaystyle G_{n} = \sum_{k=1}^{n} |f_{k}|$ and $\displaystyle G = \sum_{k=1}^{\infty} |f_{k}|$. Then
\begin{equation*}
\begin{aligned}
||G_{n}||_{p} = \big( \int \big| \sum_{k=1}^{n} |f_{k}| \big|^{p} \big)^{1/p} \\
= \big( \int (|f_{1}| + \dots + |f_{n}|)^{p} \big)^{1/p} \\
\leq \big( \int |f_{1}|^{p} + \dots + \int |f_{n}|^{p} \big)^{1/p} \\
\leq \big( \int |f_{1}|^{p} \big)^{1/p} + \dots + \big( \int |f_{n}|^{p} \big)^{1/p} \\
= \sum_{k=1}^{n} ||f_{k}||_{p} \\
\leq B
\end{aligned}
\end{equation*}
for all $n$. Therefore, by the Monotone Convergence Theorem (since $\displaystyle \sum_{k=1}^{n} |f_{k}| \leq \sum_{k=1}^{n+1}|f_{k}|)$
\begin{equation*}
\int G^{p} = \lim \int G_{n}^{p} \leq B^{p}
\end{equation*}
Thus, $G \in L^{p}$, and in particular, $G(x) < \infty$ a.e., implying that the series $\displaystyle \sum_{k=1}^{\infty} f_{k}$ converges a.e. Denoting its sum by $F$, we have $|F| \leq G$ and hence $F \in L^{p}$. Moreover,
\begin{equation*}
\begin{aligned}
|F – \sum_{k=1}^{n} f_{k}|^{p} \leq (2G)^{p} \in L^{1}
\end{aligned}
\end{equation*}
Thus, by the Dominated Convergence Theorem,
\begin{equation*}
\begin{aligned}
\lim_{n \rightarrow \infty} ||F – \sum_{k=1}^{n} f_{k}||_{p}^{p} = \lim_{n \rightarrow \infty} \int \big| F – \sum_{k=1}^{n} f_{k} \big|^{p} = \int \lim_{n \rightarrow \infty} \big| F – \sum_{k=1}^{n} f_{k} \big|^{p} = 0
\end{aligned}
\end{equation*}
Thus ,the series $\displaystyle \sum_{k=1}^{\infty} f_{k}$ converges in the $L^{p}$ norm. This completes the proof.
My question is why does this show that every absolutely convergent series in $X$ converges (thus allowing us to apply Theorem 5.1 thereby completing the proof)? Doesn't $\lim_{n \rightarrow \infty} ||F – \sum_{k=1}^{n} f_{k}||_{p}^{p} = 0$ in fact show that the series merely converges in $L^{p}$? Not converge absolutely in $L^{p}$?
Best Answer
Theorem $5.1$ says that a normed space is complete iff, whenever $\sum_{n=1}^\infty \|x_n\|$ converges, then $\sum_{n=1}^\infty x_n$ converges (in the norm, of course). Folland's wording is: "A normed vector space $X$ is complete iff every absolutely convergent series in $X$ converges." However this is the same as what I've written above, because he defines that a series $\sum_{n=1}^\infty x_n$ converges to $x$ if $\sum_{n=1}^N x_n\to x$ as $N\to\infty$, and that a series $\sum_{n=1}^\infty x_n$ is absolutely convergent if $\sum_{n=1}^\infty\|x_n\|<\infty$.
To apply this theorem, we assume that $\sum_{k=1}^\infty \|f_k\|_p$ converges and must show that $\sum_{k=1}^\infty f_k$ converges in the $L^p$ norm. This is precisely what Folland does.