| Notation | Description | Source |
|
| $E_{A}$ | Normed space associated with $A \subset E$. | Definition 13.1.14 |
| $L(E; F)$ | Continuous linear maps $E \to F$. | Definition 12.6.1 |
| $L^{n}(E_{1},\ldots,E_{n}; F)$ | Continuous $n$-linear maps $\prod E_{j} \to F$. | Definition 12.6.2 |
| $\mathfrak{B}(E)$ | Bounded subsets of TVS $E$. | Definition 12.4.1 |
| $B(T; E)$ | Bounded functions $T \to E$ with uniform topology. | Definition 12.12.2 |
| $B_{\mathfrak{S}}^{k}(E; F)$, $B(E; F)$ | $\mathfrak{S}$-bounded $k$-linear maps; bounded linear maps. | Definition 12.13.1 |
| $E^{*}$ | Topological dual of TVS $E$. | Definition 12.5.2 |
| $E_{w}$ | $E$ equipped with the weak topology. | Definition 12.5.3 |
| $\langle x, \phi \rangle_{E}$ | Duality pairing between $x \in E$ and $\phi \in E^{*}$. | Proposition 12.5.1 |
| $L_{s}(E; F)$ | $L(E; F)$ with strong operator topology. | Definition 12.13.3 |
| $L_{w}(E; F)$ | $L(E; F)$ with weak operator topology. | Definition 12.13.5 |
| $L_{b}(E; F)$ | $L(E; F)$ with topology of bounded convergence. | Definition 12.13.6 |
| $L_{c}(E; F)$ | $L(E; F)$ with topology of precompact convergence. | Definition 12.13.7 |
| $\mathcal{K}(E; F)$ | Space of compact operators from $E$ to $F$. | Definition 14.8.1 |
| $\widehat{E}$ | Hausdorff completion of TVS $E$. | Definition 12.8.1 |
| $\mathrm{Conv}(A)$ | Convex hull of $A$. | Definition 13.1.2 |
| $\aconv(A)$ | Convex circled hull of $A$. | Definition 13.1.3 |
| $[\cdot]_{A}$ | Gauge of a radial set $A$. | Definition 13.1.11 |
| $\rho_{M}$ | Quotient of seminorm $\rho$ by subspace $M$. | Definition 13.6.1 |
| $E \otimes_{\pi} F$ | Projective tensor product of $E$ and $F$. | Definition 13.11.1 |
| $E \,\wh{\otimes}_{\pi} F$ | Projective completion of $E$ and $F$. | Definition 13.11.1 |
| $p \otimes q$ | Cross seminorm of $p$ and $q$. | Definition 13.11.3 |
| $N(E; F)$ | Nuclear mappings from $E$ to $F$. | Definition 13.13.1 |
| $I(E, F)$ | Integral bilinear forms on $E \times F$. | Definition 13.12.5 |
| $x \vee y$, $x \wedge y$ | $\sup$ and $\inf$ in vector lattice. | Definition 17.3.1 |
| $|x|$ | Absolute value $x \vee (-x)$ in a vector lattice. | Definition 17.3.2 |
| $x \perp y$ | Disjointness $|x| \wedge |y| = 0$ in a vector lattice. | Definition 17.3.4 |
| $[x, y]$ | Order interval $\{z \mid x \le z \le y\}$. | Definition 17.1.4 |
| $E^{b}$ | Order bounded dual of ordered vector space $E$. | Definition 17.2.1 |
| $E^{+}$ | Order dual of $E$. | Definition 17.2.2 |
| $f^{+}$, $f^{-}$ | Positive and negative parts $f \vee 0$ and $-(f \wedge 0)$. | Definition 27.3.2 |
| $\mathscr{P}([a,b])$ | Set of all partitions of $[a,b]$. | Definition 15.1.1 |
| $\mathscr{P}_{t}([a,b])$ | Set of all tagged partitions of $[a,b]$. | Definition 15.1.2 |
| $\sigma(P)$ | Mesh of a partition $P$. | Definition 15.1.3 |
| $V_{\rho,P}(f)$ | Variation of $f$ w.r.t. seminorm $\rho$ and partition $P$. | Definition 15.2.1 |
| $[f]_{\mathrm{var},\rho}$ | Total variation of $f$ w.r.t. $\rho$. | Definition 15.2.1 |
| $T_{f,\rho}(x)$ | Variation function of $f$ with respect to $\rho$. | Definition 15.2.2 |
| $BV([a,b]; E)$ | Functions of bounded variation. | Definition 15.2.3 |
| $S(P, c, f, G)$ | Riemann-Stieltjes sum $\sum_{j} f(c_{j})[G(x_{j})-G(x_{j-1})]$. | Definition 15.3.1 |
| $\int_{a}^{b} f dG$, $\int_{a}^{b} f(t) G(dt)$ | Riemann-Stieljes integral of $f$ with respect to $G$. | Definition 15.3.2 |
| $RS([a,b], G)$ | Space of RS-integrable functions w.r.t. $G$. | Definition 15.3.2 |
| $\mathrm{Reg}([a,b], G; E)$ | Regulated functions w.r.t. $G$ on $[a,b]$. | Definition 15.6.3 |
| $\mu_{G}$ | Lebesgue-Stieltjes measure associated with $G$. | Definition 15.7.1 |
| $\int_{\gamma} f$, $\int_{\gamma} f(z)dz$ | Path integral of $f$ with respect to $\gamma$. | Definition 15.5.2 |
| $PI([a, b], \gamma; E)$ | Space of path integrable functions with respect to $\gamma$. | Definition 15.5.2 |
| $\partial f(x)$ | Subdifferential of $f$ at $x$. | Definition 19.2.1 |
| $(\phi, \alpha) \le f$ | $\phi - \alpha \le f$. $(\phi, \alpha)$ is an affine minorant of $f$. | Definition 19.3.1 |
| $f^{*}$ | Conjugate function of $f$. | Definition 19.3.2 |
| $I_{A}$ | Indicator/infinity characteristic function of $A$. | Definition 19.4.2 |
| $H_{A}$ | Support function of $A$ | Definition 19.4.1 |
| $\catc_{1}$ | Category of compatible couples in $\catc$. | Definition 20.1.4 |
| $\cf(E_{0}, E_{1})$ | Calderón space of $(E_{0}, E_{1})$ | Definition 20.2.1 |
| $[E_{0}, E_{1}]_{\theta}$ | Complex interpolation space of exponent $\theta$ for the couple $(E_{0}, E_{1})$. | Definition 20.2.2 |
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