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Added notation pages for major sections.

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Bokuan Li
2026-04-23 02:56:47 -04:00
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\chapter{Notations}
\label{chap:fa-notations}
\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 ----
$L(E; F)$ & 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} \\
$B(T; E)$ & Bounded functions $T \to E$ with uniform topology. & \autoref{definition:bounded-function-space} \\
$B_\mathfrak{S}^k(E; F)$, $B(E; F)$ & $\mathfrak{S}$-bounded $k$-linear maps; bounded linear maps. & \autoref{definition:bounded-linear-map-space} \\
$E^*$ & Topological dual of TVS $E$. & \autoref{definition:topological-dual} \\
$E_w$ & $E$ equipped with the weak topology. & \autoref{definition:weak-topology} \\
$\langle x, \phi \rangle_E$ & Duality pairing between $x \in E$ and $\phi \in E^*$. & \autoref{proposition:polarisation-linear} \\
$L_s(E; F)$ & $L(E; F)$ with strong operator topology. & \autoref{definition:strong-operator-topology} \\
$L_w(E; F)$ & $L(E; F)$ with weak operator topology. & \autoref{definition:weak-operator-topology} \\
$L_b(E; F)$ & $L(E; F)$ with topology of bounded convergence. & \autoref{definition:bounded-convergence-topology} \\
$\widehat{E}$ & Hausdorff completion of TVS $E$. & \autoref{definition:tvs-completion} \\
% ---- Locally Convex ----
$\mathrm{Conv}(A)$ & Convex hull of $A$. & \autoref{definition:convex-hull} \\
$\Gamma(A)$ & Convex circled hull of $A$. & \autoref{definition:convex-circled-hull} \\
$[\cdot]_A$ & Gauge of a radial set $A$. & \autoref{definition:gauge} \\
$\rho_M$ & Quotient of seminorm $\rho$ by subspace $M$. & \autoref{definition:quotient-norm} \\
$E \otimes_\pi F$ & Projective tensor product of $E$ and $F$. & \autoref{definition:projective-tensor-product} \\
$E \,\widetilde{\otimes}_\pi F$ & Projective completion of $E$ and $F$. & \autoref{definition:projective-tensor-product} \\
$p \otimes q$ & Cross seminorm of $p$ and $q$. & \autoref{definition:cross-seminorm} \\
% ---- Order Structures ----
$x \vee y$, $x \wedge y$ & $\sup$ and $\inf$ in vector lattice. & \autoref{definition:vector-lattice} \\
$|x|$ & Absolute value $x \vee (-x)$ in a vector lattice. & \autoref{definition:order-absolute-value} \\
$x \perp y$ & Disjointness $|x| \wedge |y| = 0$ in a vector lattice. & \autoref{definition:order-disjoint} \\
$[x, y]$ & Order interval $\{z \mid x \le z \le y\}$. & \autoref{definition:ordered-vector-space-interval} \\
$E^b$ & 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} \\
\end{tabular}