diff --git a/src/fa/notation.tex b/src/fa/notation.tex index 7a1f2ac..19ada12 100644 --- a/src/fa/notation.tex +++ b/src/fa/notation.tex @@ -4,19 +4,8 @@ \begin{tabular}{lll} \textbf{Notation} & \textbf{Description} & \textbf{Source} \\ \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 ---- + $E_A$ & 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} \\ $L^n(E_1,\ldots,E_n; F)$ & Continuous $n$-linear maps $\prod E_j \to F$. & \autoref{definition:continuous-multilinear} \\ $B(E)$ & Bounded subsets of TVS $E$. & \autoref{definition:bounded} \\ @@ -45,5 +34,17 @@ $E^b$ & Order bounded dual of ordered vector space $E$. & \autoref{definition:order-bounded-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} \\ + % ---- 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}