\chapter{Notations}
\label{chap:fa-notations}

\begin{tabular}{lll}
    \textbf{Notation} & \textbf{Description} & \textbf{Source} \\
    \hline
    % ---- 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} \\
    $B(T; E)$ & 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} \\
    % ---- 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}

