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Updated $L^p$ notations.

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Bokuan Li
2026-08-14 14:29:06 -04:00
parent e2eeae3f36
commit f8b61cca1a
2 changed files with 19 additions and 3 deletions
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14
src/fa/lp/definition.tex
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@@ -1,6 +1,12 @@
\section{Basic Properties}
\label{section:lp-basic}
\begin{definition}[$B^\infty$ Spaces]
\label{definition:bounded-borel-function}
Let $(X, \cm, \mu)$ be a measure space and $E$ be a normed vector space, then the set $B^\infty(X; E)$ is the \textbf{space of bounded $E$-valued strongly measurable functions} on $X$, and the set $B^\infty(X)$ is the space of bounded complex-valued Borel measurable functions on $X$.
\end{definition}
\begin{definition}[$\mathcal{L}^p$ Spaces]
\label{definition:lp-unequivalence}
Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, $f: X \to E$ be strongly measurable, and $p \in [1, \infty)$, then $f$ is \textbf{$p$-integrable} if
@@ -8,17 +14,19 @@
\norm{f}_{L^p(X; E)} = \norm{f}_{L^p(\mu; E)} = \norm{f}_{L^p(X, \cm, \mu; E)} = \braks{\int \norm{f}_E^p d\mu}^{1/p} < \infty
\]
The set $\mathcal{L}^p(X; E) = \mathcal{L}^p(\mu; E) = \mathcal{L}^p(X, \cm, \mu; E)$ is the space of all $p$-integrable functions on $X$.
The set $\mathcal{L}^p(X; E) = \mathcal{L}^p(\mu; E) = \mathcal{L}^p(X, \cm, \mu; E)$ is the space of all $E$-valued $p$-integrable functions on $X$.
\end{definition}
\begin{definition}[Essential Supremum]
\label{definition:esssup}
Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, and $f: X \to E$ be strongly measurable, then $f$ is \textbf{essentially bounded} if
\[
\norm{f}_{L^\infty(X; E)} = \norm{f}_{L^\infty(\mu; E)} = \norm{f}_{L^\infty(X, \cm, \mu; E)} = \inf\bracs{\alpha \ge 0|\mu(\bracs{f > \alpha}) = 0} < \infty
\norm{f}_{\mathcal{L}^\infty(X; E)} = \norm{f}_{\mathcal{L}^\infty(\mu; E)} = \norm{f}_{\mathcal{L}^\infty(X, \cm, \mu; E)} = \inf\bracs{\alpha \ge 0|\mu(\bracs{f > \alpha}) = 0} < \infty
\]
In which case, $\norm{f}_{L^\infty(X; E)}$ is the \textbf{essential supremum} of $f$.
In which case, $\norm{f}_{\mathcal{L}^\infty(X; E)}$ is the \textbf{essential supremum} of $f$.
The set $\mathcal{L}^\infty(X; E) = \mathcal{L}^\infty(\mu; E) = \mathcal{L}^\infty(X, \cm, \mu; E)$ is the space of all $E$-valued essentially bounded functions on $X$.
\end{definition}
\begin{definition}[Hölder conjugates]

8
src/topology/notation.tex
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@@ -26,6 +26,14 @@
$f \prec U$ & $f \in C_c(X; [0,1])$ with $\mathrm{supp}(f) \subset U$. & \autoref{definition:compactly-supported-01} \\
$C_0(X; E)$ & Continuous functions vanishing at infinity. & \autoref{definition:vanish-at-infinity} \\
$BC(X; E)$ & Bounded continuous functions $X \to E$. & \autoref{definition:bounded-continuous-function-space} \\
% ---- $L^p$ Spaces ----
$B^\infty(X; E)$ & Bounded $E$-valued strongly measurable functions on $X$. & \autoref{definition:bounded-borel-function} \\
$B^\infty(X)$ & Bounded $\complex$-valued Borel measurable functions on $X$. & \autoref{definition:bounded-borel-function} \\
$\mathcal{L}^p(X; E)$, $\mathcal{L}^p(\mu; E)$, $\mathcal{L}^p(X, \cm, \mu; E)$ & $E$-valued $p$-integrable functions on $X$. & \autoref{definition:lp-unequivalence} \\
$\norm{f}_{L^p(X; E)}$ & $L^p$ norm of $f$: $\braks{\int \norm{f}_E^p d\mu}^{1/p}$. & \autoref{definition:lp-unequivalence} \\
$\mathcal{L}^\infty(X; E)$, $\mathcal{L}^\infty(\mu; E)$, $\mathcal{L}^\infty(X, \cm, \mu; E)$ & $E$-valued essentially bounded functions on $X$. & \autoref{definition:esssup} \\
$\norm{f}_{\mathcal{L}^\infty(X; E)}$ & Essential supremum of $f$. & \autoref{definition:esssup} \\
$L^p(X, \cm, \mu; E)$ & $E$-valued $L^p$ space on $(X,\cm,\mu)$; quotient of $\mathcal{L}^p$ by a.e.-equality. & \autoref{definition:lp} \\
% DST
$\mathscr{N}$ & The Baire space. & \autoref{definition:the-baire-space} \\