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Added associated space notation.
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Bokuan Li
2026-05-02 16:00:33 -04:00
parent 1e53581113
commit ba5830d5b6
1 changed files with 13 additions and 12 deletions
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src/fa/notation.tex
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@@ -4,19 +4,8 @@
\begin{tabular}{lll} \begin{tabular}{lll}
\textbf{Notation} & \textbf{Description} & \textbf{Source} \\ \textbf{Notation} & \textbf{Description} & \textbf{Source} \\
\hline \hline
% ---- Riemann--Stieltjes ----
$\mathscr{P}([a,b])$ & Set of all partitions of $[a,b]$. & \autoref{definition:partition-interval} \\
$\mathscr{P}_t([a,b])$ & Set of all tagged partitions of $[a,b]$. & \autoref{definition:tagged-partition} \\
$\sigma(P)$ & Mesh of a partition $P$. & \autoref{definition:mesh} \\
$V_{\rho,P}(f)$ & Variation of $f$ w.r.t.\ seminorm $\rho$ and partition $P$. & \autoref{definition:total-variation} \\
$[f]_{\mathrm{var},\rho}$ & Total variation of $f$ w.r.t.\ $\rho$. & \autoref{definition:total-variation} \\
$T_{f,\rho}(x)$ & Variation function of $f$ with respect to $\rho$. & \autoref{definition:variation-function} \\
$BV([a,b]; E)$ & Functions of bounded variation. & \autoref{definition:bounded-variation} \\
$S(P, c, f, G)$ & Riemann-Stieltjes sum $\sum_j f(c_j)[G(x_j)-G(x_{j-1})]$. & \autoref{definition:rs-sum} \\
$RS([a,b], G)$ & Space of RS-integrable functions w.r.t.\ $G$. & \autoref{definition:rs-integral} \\
$\mathrm{Reg}([a,b], G; E)$ & Regulated functions w.r.t.\ $G$ on $[a,b]$. & \autoref{definition:regulated-function} \\
$\mu_G$ & Lebesgue-Stieltjes measure associated with $G$. & \autoref{definition:riemann-lebesgue-stieltjes} \\
% ---- Topological Vector Spaces ---- % ---- Topological Vector Spaces ----
$E_A$ & Normed space associated with $A \subset E$. & \autoref{definition:lc-associated-normed-space} \\
$L(E; F)$ & Continuous linear maps $E \to F$. & \autoref{definition:continuous-linear} \\ $L(E; F)$ & Continuous linear maps $E \to F$. & \autoref{definition:continuous-linear} \\
$L^n(E_1,\ldots,E_n; F)$ & Continuous $n$-linear maps $\prod E_j \to F$. & \autoref{definition:continuous-multilinear} \\ $L^n(E_1,\ldots,E_n; F)$ & Continuous $n$-linear maps $\prod E_j \to F$. & \autoref{definition:continuous-multilinear} \\
$B(E)$ & Bounded subsets of TVS $E$. & \autoref{definition:bounded} \\ $B(E)$ & Bounded subsets of TVS $E$. & \autoref{definition:bounded} \\
@@ -45,5 +34,17 @@
$E^b$ & Order bounded dual of ordered vector space $E$. & \autoref{definition:order-bounded-dual} \\ $E^b$ & Order bounded dual of ordered vector space $E$. & \autoref{definition:order-bounded-dual} \\
$E^+$ & Order dual of $E$. & \autoref{definition:order-dual} \\ $E^+$ & Order dual of $E$. & \autoref{definition:order-dual} \\
$f^+$, $f^-$ & Positive and negative parts $f \vee 0$ and $-(f \wedge 0)$. & \autoref{definition:positive-negative-parts} \\ $f^+$, $f^-$ & Positive and negative parts $f \vee 0$ and $-(f \wedge 0)$. & \autoref{definition:positive-negative-parts} \\
% ---- Riemann--Stieltjes ----
$\mathscr{P}([a,b])$ & Set of all partitions of $[a,b]$. & \autoref{definition:partition-interval} \\
$\mathscr{P}_t([a,b])$ & Set of all tagged partitions of $[a,b]$. & \autoref{definition:tagged-partition} \\
$\sigma(P)$ & Mesh of a partition $P$. & \autoref{definition:mesh} \\
$V_{\rho,P}(f)$ & Variation of $f$ w.r.t.\ seminorm $\rho$ and partition $P$. & \autoref{definition:total-variation} \\
$[f]_{\mathrm{var},\rho}$ & Total variation of $f$ w.r.t.\ $\rho$. & \autoref{definition:total-variation} \\
$T_{f,\rho}(x)$ & Variation function of $f$ with respect to $\rho$. & \autoref{definition:variation-function} \\
$BV([a,b]; E)$ & Functions of bounded variation. & \autoref{definition:bounded-variation} \\
$S(P, c, f, G)$ & Riemann-Stieltjes sum $\sum_j f(c_j)[G(x_j)-G(x_{j-1})]$. & \autoref{definition:rs-sum} \\
$RS([a,b], G)$ & Space of RS-integrable functions w.r.t.\ $G$. & \autoref{definition:rs-integral} \\
$\mathrm{Reg}([a,b], G; E)$ & Regulated functions w.r.t.\ $G$ on $[a,b]$. & \autoref{definition:regulated-function} \\
$\mu_G$ & Lebesgue-Stieltjes measure associated with $G$. & \autoref{definition:riemann-lebesgue-stieltjes} \\
\end{tabular} \end{tabular}