Examples:: *[[Examples]]* Nonexamples:: [[L0 norm]] Constructions::  Specializations:: [[norm|Euclidean Norm]], [[taxicab norm]] Generalizations:: [[Lp space]]s Justifications and Intuition:: *[[Justifications and Intuition]]* ---- Throughout, $\mathbb{F}$ denotes $\mathbb{R}$ or $\mathbb{C}$. > [!definition] Definition. ([[Lp-norm]] for [[measure|measure spaces]]) > Let $(X,\Sigma, \mu)$ be a [[measure|measure space]], $1\leq p<\infty$, and suppose $f:X \to \mathbb{F}$ is $\Sigma$-[[measurable function|measurable]]. The **$L^{p}$-norm** of $f$ is obtained by exponentiating the $p$th [[moment of a measurable function|absolute moment]] of $f$ so as to make homogeneity hold, namely, $\|f\|_{p}:=\left( \int |f|^{p} \, d\mu \right)^{1/p}.$ > Also, $\|f\|_{\infty}$, which is called the **essential supremum** of $f$, is defined[^1] $\|f\|_{\infty}:= \inf\{ t>0: \mu(\{ x \in X: |f(x)|>t \})=0 \}.$ > > > > We define the [[vector space]] $\mathcal{L}^{p}(X, \Sigma, \mu):=\mathcal{L}^{p}(\mu):=\{ f:X \to \mathbb{F} \text{ measurable}: \|f\|_{p}<\infty \}.$ > Importantly, $\|\cdot\|_{p}$ is merely a [[seminorm]] on $\mathcal{L}^{p}(\mu)$: it is not [[injective sheaf morphism|injective]] because for any $f$ satisfying $\mu\big( f ^{-1}(\mathbb{F}- \{ 0 \})\big)=0$ we have $\|f\|_p=0$. $\|\cdot\|_{p}$ descends to a [[norm]][^8] once we identify functions that agree [[almost-everywhere]]: define the **$L^{p}$ space** as the [[quotient module|quotient vector space]] > > $L^{p}(X, \Sigma, \mu):= \frac{\mathcal{L}^{p}(\mu)}{\operatorname{ker }\big(\|\cdot\|_{p}:\mathcal{L}^{p}(\mu) \to \mathbb{F}\big)}=\frac{\mathcal{L}^{p}(\mu)}{\{ f \sim g \iff f=g \ \text{a.e.} \}},$ > whence the [[universal property]] of [[quotient set|of quotients]] gives a diagram > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYABkAvgD00ACk5cmAShBjS6TLnyEUARnJVajFmzm8+AIzPAAYmOWqQGbHgJEy2-fWatEIEVNncisr6MFAA5vBEoABmAE4QXEhkIDgQSLoGXsbsAD4ABJwCUBA4Bbl2MfGJiMmpSABMKpUJDdR1iBkMWGDeIFB0cHyhINSeRj6cMAAeWHA4cHkAhGUA7liweAywwJz5hcWlu7bUDHRmMAwACurOWiCxWGF8OCMpdFgMbHwQEADWwWIgA > \begin{tikzcd} > \mathcal{L}^p(\mu) \arrow[r, "\| \cdot \|_p"] \arrow[d] & \mathbb{F} \\ > L^p(\mu) \arrow[ru, "\exists ! \widetilde{\| \cdot \|_p}"', dashed, hook] > > \end{tikzcd} > \end{document} > ``` > > In practice, we just denote the induced [[norm]] $L^{p}(\mu) \to \mathbb{F}$ as $\|\cdot\|_p$ instead of $\widetilde{\|\cdot\|_{p}}$, thinking about elements of $L^{p}(\mu)$ as functions rather than [[coset|classes]] thereof. This fiction is harmless provided the operations you perform with such 'functions' produce the same results if the functions are changed on a set of [[measure zero]]. > [[finite measure|When]] $\mu(X)<\infty$, [[antimonotone inclusion of Lebesgue spaces wrt a finite measure|we have the antimonotonicity]] $p < q \implies \mathcal{L}^{p}(\mu) \supset \mathcal{L}^{q}(\mu)$ for all $p,q \in (0, \infty)$. > [!Equivalence] > We can access $\|f\|_{p}$ by 'probing with small test functions'. > > ![[converse to Hölder's inequality#^theorem]] > > This formula is useful e.g. when proving [[Minkowski's inequality]] or [[Lp duality]]. > > > > ![[converse to Hölder's inequality#^equivalence]] > > ![[converse to Hölder's inequality#^note]] > > > [!basicproperties] > - [[Lp spaces are Banach]] > > ^properties - when $\operatorname{ess}\operatorname{\sup}=\sup$ [^8]: Moreover, the [[topological space|topology]] [[metric topology|induced by]] the [[quotient set|descended]] [[norm]] is precisely the [[quotient topology]], cf. [[characterization of quotienting a Banach space]] (or something like that, going to make the note tomorrow) [^2]: The exponent $1/p$ serves to ensure homogeneity holds, as will be seen in the justification. > [!specialization] Specialization. ([[Lp-norm]] for coordinate [[vector space|vector spaces]]) > Suppose $\mu$ is the [[measure|counting measure]] on $\mathbb{N}$. If $a=(a_{1},a_{2},\dots) \in \mathbb{F}^{\mathbb{N}}$ is a [[sequence]] in $\mathbb{F}$ and $0<p<\infty$, [[integral|then]][^3] $\|a\|_{p}= (\sum_{k=1}^{\infty}|a_{k}|^{p})^{1/p}\text{ and }\|a\|_{\infty}=\sup \{ |a_{k}|: k \in \mathbb{N} \}.$ > Specializing further, this yields for a finite-dimensional coordinate vector space $\mathbb{F}^{N}$ the [[norm|norms]] $\|x\|_p = \left( \sum_{j=1}^{n}|x_{j}|^{p} \right)^{1/p}, \|x\|_{\infty}=\max_{j \in [N]} \{ |x_{j}| \}.$ > Note that $\mathcal{L}^{p}=L^{p}$ in this setting because only $\emptyset$ has counting measure zero. The notations $\ell^{p}, \ell^{\infty}$ are adopted accordingly. We call $\|x\|_{\infty}$ the **sup norm** in this setting. > [!justification] > The definition of $\|\cdot\|_{p}$ is so that homogeneity will hold (verified); together with [[Minkowski's inequality]] we have that $\mathcal{L}^{p}(\mu)$ is a [[vector space]] and $\|\cdot\|_{p}$ is a [[seminorm]] on it. ^justification [^3]:(Note that $\operatorname{ess}\sup=\sup$ for the [[measure|counting measure]] because there are no nonempty sets of [[measure zero]]). [^1]: $\|f\|_{p}$ is unchanged if $f$ is modified on a set of [[measure zero]]. By using the *essential* supremum instead of the [[supremum]] in the definition of $\|f\|_{\infty}$, we arrange for $\|\cdot\|_{\infty}$ to enjoy this same property. Think of $\|f\|_{\infty}$ as the smallest you can make $\sup |f|$ after modifications on sets of [[measure zero]]. ---- #### ---- #### 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 > ``` > [!definition] Definition. ([[Lp-norm]] for f.d. [[vector space|vector spaces]]) > Let $V$ be a finite-dimensional [[vector space]] over $\mathbb{F}$. > > For $0<p<\infty$ , the function $\|\cdot\|_p : V \times V \to \rr$ given by $\|x\|_p = \left( \sum_{j=1}^{n}|x_{j}|^{p} \right)^{1/p}$ is a [[norm]] on $V$, called the $L^{p}$**-norm**. \ The **sup norm** is defined as $\|x\|_{\infty}:=\sup \{ |x_{1}|, |x_{2}|,\dots \}=\lim_{ p \to \infty }\|x\|_{p}$. \ For $V$ finite dimensional the **counting measure** or **0-norm** is $\|x\|_{0}:=\lim_{ p \to 0 }\|x\|_{p}^{p}=\sum_{i}^{}\mathbb{1}_{\{ x_{i} \neq 0 \}_{}}$. > [!warning] > The **counting measure** is NOT a [[norm]]! > [!equivalence] Equivalence. ("Converse to [[Hölder's inequality]]") > An application of [[Hölder's inequality]] shows that $\|f\|_{p}$ is equivalently given by the 'probe with small functions' formula $\|f\|_{p}=\sup \left\{ \left|\int fh \, d\mu \right| : h \in \mathcal{L}^{p'}(\mu) \text{ and }\|h\|_{p'} \leq 1 \right\}.$ This formula is useful e.g. when proving [[Minkowski's inequality]]. > > > > [!proof]- Proof of Equivalence. > > If $\|f\|_{p}=0$ [[dual exponent|then]] by the [[triangle inequality for integrals]] and [[Hölder's inequality]] $|\int fh \, d\mu| \leq \int |fh| \, d\mu \leq \|f\|_{p} \|h\|_{p'}=0$ > > and so the result holds. So assume $\|f\|_{p} \neq 0$. > > > > [[Hölder's inequality]] implies that if $h \in \mathcal{L}^{p'}(\mu)$ with $\|h\|_{p'} \leq 1$, then $\begin{align} > > |\int fh \, d\mu | \leq \int |fh| \, d\mu \leq \|f\|_{p} \|h\|_{p'} \leq \|f\|_{p} > > \end{align}$ > > from which RHS $\leq$ LHS follows. For the reverse inequality, it suffices to find $h \in \mathcal{L}^{p'}(\mu)$, $\|h\|_{p'} \leq 1$, such that $\|f\|_{p} \leq |\int fh \, d\mu|$. To this end, define $h(x)=\begin{cases} > > \frac{\overline{f(x)}|f(x)|^{p-2}}{\|f\|_{p}^{p/p'}} & f(x) \neq 0 \\ > > 0 & f(x)=0. > > \end{cases}$ > > Then $\|h\|_{p'} =1$ > > (showed on paper, should bring over here later) > > > > and (using $f(x) \overline{f(x)}=|f(x)|^{2}$) $\begin{align} > > \int fh \, d\mu &= \int_{x \in X} \frac{f(x) \overline{f(x)} |f(x)|^{p-2}}{\|f\|_{p}^{p/p'}} \, d\mu \\ > > &= \frac{1}{\|f\|_{p}^{p/p'}} \int _{x \in X} {|f(x)|^{p}}{} \, d\mu \\ > > &= \frac{\|f\|_{p}^{p}}{\|f\|_{p}^{p/p'}}\\ > > &= \|f\|_{_{p}}, > > \end{align}$ > > where the last step [[dual exponent|used that]] $p-\frac{p}{p'}=1$. > > > > >