| Notation | Description | Source |
|
| $\mathscr{P}([a,b])$ | Set of all partitions of $[a,b]$. | Definition 13.1.1 |
| $\mathscr{P}_{t}([a,b])$ | Set of all tagged partitions of $[a,b]$. | Definition 13.1.2 |
| $\sigma(P)$ | Mesh of a partition $P$. | Definition 13.1.3 |
| $V_{\rho,P}(f)$ | Variation of $f$ w.r.t. seminorm $\rho$ and partition $P$. | Definition 13.2.1 |
| $[f]_{\mathrm{var},\rho}$ | Total variation of $f$ w.r.t. $\rho$. | Definition 13.2.1 |
| $T_{f,\rho}(x)$ | Variation function of $f$ with respect to $\rho$. | Definition 13.2.2 |
| $BV([a,b]; E)$ | Functions of bounded variation. | Definition 13.2.3 |
| $S(P, c, f, G)$ | Riemann-Stieltjes sum $\sum_{j} f(c_{j})[G(x_{j})-G(x_{j-1})]$. | Definition 13.3.1 |
| $RS([a,b], G)$ | Space of RS-integrable functions w.r.t. $G$. | Definition 13.3.2 |
| $\mathrm{Reg}([a,b], G; E)$ | Regulated functions w.r.t. $G$ on $[a,b]$. | Definition 13.5.3 |
| $\mu_{G}$ | Lebesgue-Stieltjes measure associated with $G$. | Definition 13.6.1 |
| $L(E; F)$ | Continuous linear maps $E \to F$. | Definition 10.5.1 |
| $L^{n}(E_{1},\ldots,E_{n}; F)$ | Continuous $n$-linear maps $\prod E_{j} \to F$. | Definition 10.5.2 |
| $B(E)$ | Bounded subsets of TVS $E$. | Definition 10.3.1 |
| $B(T; E)$ | Bounded functions $T \to E$ with uniform topology. | Definition 10.11.2 |
| $B_{\mathfrak{S}}^{k}(E; F)$, $B(E; F)$ | $\mathfrak{S}$-bounded $k$-linear maps; bounded linear maps. | Definition 10.11.4 |
| $E^{*}$ | Topological dual of TVS $E$. | Definition 10.4.2 |
| $E_{w}$ | $E$ equipped with the weak topology. | Definition 10.4.3 |
| $\langle x, \phi \rangle_{E}$ | Duality pairing between $x \in E$ and $\phi \in E^{*}$. | Proposition 10.4.1 |
| $L_{s}(E; F)$ | $L(E; F)$ with strong operator topology. | Definition 10.11.6 |
| $L_{w}(E; F)$ | $L(E; F)$ with weak operator topology. | Definition 10.11.8 |
| $L_{b}(E; F)$ | $L(E; F)$ with topology of bounded convergence. | Definition 10.11.9 |
| $\widehat{E}$ | Hausdorff completion of TVS $E$. | Definition 10.7.1 |
| $\mathrm{Conv}(A)$ | Convex hull of $A$. | Definition 11.1.2 |
| $\Gamma(A)$ | Convex circled hull of $A$. | Definition 11.1.3 |
| $[\cdot]_{A}$ | Gauge of a radial set $A$. | Definition 11.1.11 |
| $\rho_{M}$ | Quotient of seminorm $\rho$ by subspace $M$. | Definition 11.4.1 |
| $E \otimes_{\pi} F$ | Projective tensor product of $E$ and $F$. | Definition 11.9.1 |
| $E \,\widetilde{\otimes}_{\pi} F$ | Projective completion of $E$ and $F$. | Definition 11.9.1 |
| $p \otimes q$ | Cross seminorm of $p$ and $q$. | Definition 11.9.2 |
| $x \vee y$, $x \wedge y$ | $\sup$ and $\inf$ in vector lattice. | Definition 15.1.8 |
| $|x|$ | Absolute value $x \vee (-x)$ in a vector lattice. | Definition 15.1.9 |
| $x \perp y$ | Disjointness $|x| \wedge |y| = 0$ in a vector lattice. | Definition 15.1.11 |
| $[x, y]$ | Order interval $\{z \mid x \le z \le y\}$. | Definition 15.1.3 |
| $E^{b}$ | Order bounded dual of ordered vector space $E$. | Definition 15.1.6 |
| $E^{+}$ | Order dual of $E$. | Definition 15.1.7 |
| $f^{+}$, $f^{-}$ | Positive and negative parts $f \vee 0$ and $-(f \wedge 0)$. | Definition 22.3.2 |