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[parent] proof that $L^p$ spaces are complete (Proof)

Let's prove completeness for the classical Banach spaces, say $ L^p[0,1]$ where $ p \geq 1$.

Since the case $ p=\infty$ is elementary, we may assume $ 1 \le p < \infty$. Let $ [f_{\cdot}] \in (L^p)^{\mathbf{N}}$ be a Cauchy sequence. Define $ [g_0] := [f_0]$ and for $ n > 0$ define $ [g_n] := [f_n - f_{n-1}]$. Then $ [\sum_{n=0}^N g_n] = [f_N]$ and we see that

$\displaystyle \sum_{n=0}^\infty \Vert g_n\Vert = \sum_{n=0}^\infty \Vert f_n - f_{n-1}\Vert \leq ??? < \infty.$
Thus it suffices to prove that etc.

It suffices to prove that each absolutely summable series in $ L^p$ is summable in $ L^p$ to some element in $ L^p$.

Let $ \{f_n\}$ be a sequence in $ L^p$ with $ \sum_{n=1}^\infty \Vert f_n\Vert=M<\infty$, and define functions $ g_n$ by setting $ g_n(x)=\sum_{k=1}^n\vert f_k(x)\vert$. From the Minkowski inequality we have

$\displaystyle \Vert g_n\Vert\le\sum_{k=1}^n\Vert f_k\Vert\le M. $
Hence
$\displaystyle \int g_n^p\le M^p. $
For each $ x$, $ \{g_n(x)\}$ is an increasing sequence of (extended) real numbers and so must converge to an extended real number $ g(x)$. The function $ g$ so defined is measurable, and, since $ g_n\ge 0$, we have
$\displaystyle \int g^p\le M^p $
by Fatou's Lemma. Hence $ g^p$ is integrable, and $ g(x)$ is finite for almost all $ x$.

For each $ x$ such that $ g(x)$ is finite the series $ \sum_{k=1}^\infty f_k(x)$ is an absolutely summable series of real numbers and so must be summable to a real number $ s(x)$. If we set $ s(x)=0$ for those $ x$ where $ g(x)=\infty$, we have defined a function $ s$ which is the limit almost everywhere of the partial sums $ s_n=\sum_{k=1}^n f_k$. Hence $ s$ is measurable. Since $ \vert s_n(x)\vert\le g(x)$, we have $ \vert s(x)\vert\le g(x)$. Consequently, $ s$ is in $ L^p$ and we have

$\displaystyle \vert s_n(x)-s(x)\vert^p\le 2^p\,[g(x)]^p. $
Since $ 2^pg^p$ is integrable and $ \vert s_n(x)-s(x)\vert^p$ converges to 0 for almost all $ x$, we have
$\displaystyle \int\vert s_n-s\vert^p\to 0 $
by the Lebesgue Convergence Theorem. Thus $ \Vert s_n-s\Vert^p\to 0$, whence $ \Vert s_n-s\Vert\to 0$. Consequently, the series $ \{f_n\}$ has in $ L^p$ the sum $ s$.

Bibliography

Royden, H. L. Real analysis. Third edition. Macmillan Publishing Company, New York, 1988.



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Cross-references: sum, partial sums, limit, almost all, finite, Fatou's lemma, measurable, extended real number, converge, real numbers, increasing, Minkowski inequality, functions, sequence, series, Cauchy sequence, Banach spaces
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This is version 5 of proof that $L^p$ spaces are complete, born on 2004-10-02, modified 2006-10-13.
Object id is 6270, canonical name is ProofThatLpSpacesAreComplete.
Accessed 7217 times total.

Classification:
AMS MSC: 46B25 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Classical Banach spaces in the general theory)
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