----- > [!proposition] Proposition. ([[Lp spaces are Banach]]) > Let $(X, \Sigma, \mu)$ be a [[measure|measure space]], $1 \leq p \leq \infty$. [[Lp-norm|The]] [[norm|normed]] [[vector space]] $L^{p}(\mu)=\mathcal{L}^{p}(\mu)/{\sim}$ is [[complete]], making it a [[Banach space]]. ^proposition A sequence can converge in $p$-norm without converging pointwise anywhere. However, the argument that $L^{p}$ spaces are Banach provided far below incidentally proves that there is guaranteed to be a *subsequence* converging pointwise almost everywhere. This result is of independent interest and so we extract it here: > [!proposition] Proposition. ([[sequence|convergent]] [[sequence|sequences]] in $\mathcal{L}^{p}$ have [[pointwise converge|pointwise convergent]] [[subsequence|subsequences]]) > Suppose $(X,\Sigma, \mu)$ is a [[measure|measure space]] and $1 \leq p \leq \infty$. Suppose $f \in \mathcal{L}^{p}(\mu)$ and $f_{1},f_{2},\dots$ is a [[sequence]] of functions in $\mathcal{L}^{p}(\mu)$ such that $\lim_{k \to \infty}\|f_{k}-f\|_{p}=0$. Then there exists a [[subsequence]] $f_{k_{1}}, f_{k_{2}},\dots$ such that $\lim_{m \to \infty }f_{k_{m}}(x) = f(x)$ > for [[almost-everywhere|almost every]] $x \in X$. > > > ^proposition > [!proof]- Proof. ([[Lp spaces are Banach]]) > ~ (The proof was covered in A4 integration, so it's not examinable here.) It is enough to show the following lemma which in turn makes use of results like [[Cauchy sequence is convergent iff has convergent subsequence]], [[Minkowski's inequality]], [[monotone convergence theorem for nonnegative measurable functions|MCT]], and [[Fatou's Lemma]]. > [!proposition] Lemma. ([[Cauchy sequence|Cauchy sequences]] in $\mathcal{L}^{p}(\mu)$ [[sequence|converge]]) > Suppose $(X, \Sigma, \mu)$ is a [[measure|measure space]] and $1 \leq p \leq \infty$. Then any [[Cauchy sequence]] in $\mathcal{L}^{p}(X,\Sigma, \mu)$ [[sequence|converges]]. > ^proposition > [!proof] Proof of Lemma. > ^proof [[Cauchy sequence is convergent iff has convergent subsequence|It suffices to show]] $\lim_{m \to \infty}\|f_{k_{m}}-f\|_{p}=0$ for some $f \in \mathcal{L}^{p}(\mu)$ and some [[subsequence]] $f_{k_{1}}, f_{k_{2}},\dots$. By the 'thinning until summable' property in [[Cauchy sequence]], by dropping to a [[subsequence]] (without relabeling) we may assume[^1] $\sum_{k=1}^{\infty}\|f_{k}-f_{k-1}\|_{p}< \infty.$ Consider the functions $g_{1},g_{2},\dots$ and $g$ from $X$ to $[0, \infty]$ defined by $g_{m}(x)=\sum_{k=1}^{m} |f_{k}(x)-f_{k-1}(x)| \text{ and } g(x)=\sum_{k=1}^{\infty}|f_{k}(x)-f_{k-1}(x)|.$ By definition, $g_{m}(x) \uparrow g(x)$ for every $x$. [[Minkowski's inequality]] implies that $\|g_{m}\|_{p} \leq \sum_{k=1}^{m} \|f_{k}-f_{k-1}\|_{p};$ together with the [[monotone convergence theorem for nonnegative measurable functions|monotone convergence theorem]] this implies $\int g^{p} \, d\mu = \lim_{m \to \infty}\int g_{m}^{p} \, d\mu \leq (\sum_{k=1}^{\infty} \|f_{k}-f_{k-1}\|_{p})^{p}<\infty \ \ (*)$ Thus $g(x)<\infty$ for almost every $x \in X$. Because [[series|absolute convergence]] implies [[series|convergence]] for [[series]] of real numbers, for almost every $x \in X$ we can define $f(x)= \sum_{k=1}^{\infty}\big( f_{k}(x)-f_{k-1}(x) \big)=\lim_{m \to \infty} \sum_{k=1}^{m} \big( f_{k}(x)-f_{k-1}(x) \big)$ which telescopes to equal $f(x)= \lim_{m \to \infty}f_{m}(x),$ (in particular, $\lim_{m \to \infty}f_{m}(x)$ exists for [[almost-everywhere|almost every]] $x \in X$). Define $f(x)=0$ for those $x \in X$ for which the limit does not exist. We now have a function $f$ that is the [[pointwise converge|pointwise limit]] ([[almost-everywhere]]) of $f_{1},f_{2},\dots$. The definition of $f$ shows $|f(x)| \leq g(x)$ for almost every $x \in X$. Hence from $(*)$ [[Lp-norm|we see]] $f \in \mathcal{L}^{p}(\mu)$. To show $\lim_{k \to \infty}\|f_{k}-f\|_p=0$, suppose $\varepsilon>0$ and let $n \in \mathbb{N}$ be such that $\|f_{j}-f_{k}\|_{p}<\varepsilon$ for all $j,k \geq n$. Suppose $k \geq n$. Then $\begin{align} \|f_{k}-f\|_{p} &= \left( \int |f_{k}-f|^{p} \, d\mu \right)^{1/p} \\ & \leq \liminf_{j \to \infty} \left( \int |f_{k}-f_{j}|^{p} \, d\mu \right)^{1/p} \\ &= \liminf_{j \to \infty} \|f_{k}-f_{j}\|_{p} \\ & \leq \varepsilon, \end{align}$ where the second line comes from [[Fatou's Lemma]]. Thus $\lim \|f_{k}-f\|_{p}=0$, as desired. ----- #### [^1]: A similar assumption was useful in the proof that [[series|absolute convergence]] implying [[series|convergence]] for [[series]] characterizes [[Banach space|Banach spaces]]. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```