diff --git a/src/cat/index.tex b/src/cat/index.tex index ad928be..3032c9a 100644 --- a/src/cat/index.tex +++ b/src/cat/index.tex @@ -4,3 +4,4 @@ \input{./cat/index.tex} \input{./gluing/index.tex} \input{./tricks/index.tex} +\input{./notation/index.tex} diff --git a/src/cat/notation/index.tex b/src/cat/notation/index.tex new file mode 100644 index 0000000..619ed65 --- /dev/null +++ b/src/cat/notation/index.tex @@ -0,0 +1,18 @@ +\chapter{Notations} +\label{chap:notations} + +\begin{tabular}{lll} + \textbf{Notation} & \textbf{Description} & \textbf{Source} \\ + \hline + % ---- Category Theory ---- + $\obj{\catc}$ & Objects of category $\catc$. & \autoref{definition:category} \\ + $\mor{A, B}$ & Morphisms from $A$ to $B$ in a category. & \autoref{definition:category} \\ + $\text{Id}_A$ & Identity morphism on $A$. & \autoref{definition:category} \\ + $E \otimes F$, $x_1 \otimes \cdots \otimes x_n$ & Tensor product of modules; image of $(x_1,\ldots,x_n)$ under canonical embedding. & \autoref{definition:tensor-product} \\ + $\lim_{\longrightarrow} A_i$ & Direct limit of an upward-directed system. & \autoref{definition:direct-limit} \\ + $\lim_{\longleftarrow} A_i$ & Inverse limit of a downward-directed system. & \autoref{definition:inverse-limit} \\ + $\mathbb{D}_n$, $\mathbb{D}$ & Dyadic rationals of level $n$; all dyadic rationals. & \autoref{definition:dyadic} \\ + $\mathrm{rk}(q)$ & Dyadic rank of $q \in \mathbb{D}$. & \autoref{definition:dyadic-rank} \\ + $M(x)$ & Unique $M(x) \subset \mathbb{N}^+ \cap [1, \mathrm{rk}(x)]$ such that $x = \sum_{n \in M(x)} 2^{-n}$. & \autoref{proposition:dyadic-subset} +\end{tabular} + diff --git a/src/dg/index.tex b/src/dg/index.tex index 9e718ea..0e5023f 100644 --- a/src/dg/index.tex +++ b/src/dg/index.tex @@ -2,3 +2,4 @@ \label{part:diffgeo} \input{./derivative/index.tex} +\input{./notation.tex} diff --git a/src/dg/notation.tex b/src/dg/notation.tex new file mode 100644 index 0000000..2dc73bc --- /dev/null +++ b/src/dg/notation.tex @@ -0,0 +1,16 @@ +\chapter{Notations} +\label{chap:dg-notations} + +Differential geometry is the study of things invariant under change of notation. + +\begin{tabular}{lll} + \textbf{Notation} & \textbf{Description} & \textbf{Source} \\ + \hline + $\mathcal{H}(E;F)$, $\mathcal{R}(E;F)$ & Space of derivatives; space of remainders in an $\mathcal{HR}$-system. & \autoref{definition:derivative-system} \\ + $D_{\mathcal{HR}} f(x_0)$ & $\mathcal{HR}$-derivative of $f$ at $x_0$. & \autoref{definition:space-differentiability} \\ + $\mathcal{R}_\sigma^n(E; F)$, $\mathcal{R}_\sigma(E;F)$ & $\sigma$-small functions of order $n$; order 1. & \autoref{definition:differentiation-small} \\ + $D_\sigma f(x_0)$ & $\sigma$-derivative of $f$ at $x_0$. & \autoref{definition:derivative-sets} \\ + $D_\sigma^n f$ & $n$-fold $\sigma$-derivative. & \autoref{definition:n-differentiable-sets} \\ + $x^{(k)}$ & Tuple of $x$ repeated $k$ times. & \autoref{proposition:multilinear-derivative} \\ + $D^+f(x)$ & Right derivative of $f$ at $x$. & \autoref{definition:right-differentiable-mvt} \\ +\end{tabular} diff --git a/src/fa/index.tex b/src/fa/index.tex index 4e4b2e1..b8c5ca5 100644 --- a/src/fa/index.tex +++ b/src/fa/index.tex @@ -8,3 +8,4 @@ \input{./rs/index.tex} \input{./lp/index.tex} \input{./order/index.tex} +\input{./notation.tex} diff --git a/src/fa/notation.tex b/src/fa/notation.tex new file mode 100644 index 0000000..7a1f2ac --- /dev/null +++ b/src/fa/notation.tex @@ -0,0 +1,49 @@ +\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} + diff --git a/src/measure/index.tex b/src/measure/index.tex index 51098d6..321fb91 100644 --- a/src/measure/index.tex +++ b/src/measure/index.tex @@ -8,3 +8,4 @@ \input{./measurable-maps/index.tex} \input{./lebesgue-integral/index.tex} \input{./bochner-integral/index.tex} +\input{./notation.tex} diff --git a/src/measure/notation.tex b/src/measure/notation.tex new file mode 100644 index 0000000..72856de --- /dev/null +++ b/src/measure/notation.tex @@ -0,0 +1,33 @@ +\chapter{Notations} +\label{chap:measure-notations} + +\begin{tabular}{lll} + \textbf{Notation} & \textbf{Description} & \textbf{Source} \\ + \hline + + $\sigma(\mathcal{E})$ & $\sigma$-algebra generated by $\mathcal{E}$. & \autoref{definition:generated-sigma-algebra} \\ + $\lambda(\mathcal{E})$ & $\lambda$-system generated by $\mathcal{E}$. & \autoref{definition:generated-lambda-system} \\ + % ---- Measure Theory ---- + $\mathcal{B}_X$ & Borel $\sigma$-algebra on $X$. & \autoref{definition:borel-sigma-algebra} \\ + $\sigma(\{f_i \mid i \in I\})$ & $\sigma$-algebra generated by the maps $\{f_i\}$. & \autoref{definition:generated-sigma-algebra-function} \\ + $\bigotimes_{i \in I} \mathcal{M}_i$ & Product $\sigma$-algebra. & \autoref{definition:product-sigma-algebra} \\ + $\chi_E = \mathbf{1}_E$ & Indicator function of $E$. & \autoref{definition:indicator-function} \\ + $\Sigma(X, \mathcal{M}; E)$ & Space of $E$-valued simple functions on $(X, \mathcal{M})$. & \autoref{definition:simple-function-standard-form} \\ + $\Sigma^+(X, \mathcal{M})$ & Space of non-negative simple functions. & \autoref{definition:simple-function-scalar} \\ + $\mathcal{L}^+(X, \mathcal{M})$ & Space of non-negative measurable functions. & \autoref{definition:measurable-non-negative} \\ + $f_*\mu$ & Pushforward of $\mu$ by $f$. & \autoref{definition:pushforward-measure} \\ + $\mu \otimes \nu$ & Product measure. & \autoref{definition:product-measure} \\ + $|\mu|$ & Total variation measure of a signed/vector measure. & \autoref{definition:total-variation-signed}, \autoref{definition:total-variation-vector} \\ + $\mu = \mu^+ - \mu^-$ & Jordan decomposition of a signed measure. & \autoref{theorem:jordan-decomposition} \\ + $\mu \perp \nu$ & Mutual singularity. & \autoref{definition:mutually-singular} \\ + $\nu \ll \mu$ & $\nu$ is absolutely continuous w.r.t. $\mu$. & \autoref{definition:absolutely-continuous} \\ + $M(X, \mathcal{M}; E)$, & Space of finite $E$-valued measures. & \autoref{definition:vector-measure-finite-space} \\ + $\|\mu\|_{\mathrm{var}}$ & Total variation of $\mu$. & \autoref{definition:vector-measure-finite-space} \\ + $M_R(X; E)$ & Space of finite Radon $E$-valued measures on $X$. & \autoref{definition:space-radon-measures} \\ + % ---- Lebesgue Integral ---- + $\mathcal{L}^p(X, \mathcal{M}, \mu; E)$ & Space of $p$-integrable functions, without quotient. & \autoref{definition:lp-unequivalence} \\ + $\|f\|_{L^p}$, & $L^p$ norm of $f$. & \autoref{definition:esssup} \\ + $L^p(X, \mathcal{M}, \mu; E)$ & Space of $p$-integrable functions, modulo equality almost everywhere. & \autoref{definition:lp} \\ + +\end{tabular} + diff --git a/src/topology/index.tex b/src/topology/index.tex index fda6552..c12bc4e 100644 --- a/src/topology/index.tex +++ b/src/topology/index.tex @@ -5,3 +5,4 @@ \input{./uniform/index.tex} \input{./functions/index.tex} \input{./metric/index.tex} +\input{./notation.tex} diff --git a/src/topology/notation.tex b/src/topology/notation.tex new file mode 100644 index 0000000..224aa6d --- /dev/null +++ b/src/topology/notation.tex @@ -0,0 +1,28 @@ +\chapter{Notations} +\label{chap:topology-notations} + +\begin{tabular}{lll} + \textbf{Notation} & \textbf{Description} & \textbf{Source} \\ + \hline + % ---- General Topology ---- + $\mathcal{N}_X(A)$, $\mathcal{N}(A)$, $\mathcal{N}^o(A)$ & Neighbourhood filter at $A$; open neighbourhoods of $A$. & \autoref{definition:neighbourhood} \\ + $C(X; Y)$ & Continuous functions $X \to Y$. & \autoref{definition:continuity} \\ + $E(d, r)$ & $\{(x,y) \in X \times X \mid d(x,y) < r\}$ for pseudometric $d$. & \autoref{definition:pseudometric-uniformity} \\ + $B(x, r)$ & Open ball $\{y \in X \mid d(x,y) < r\}$ for pseudometric $d$. & \autoref{definition:pseudometric-uniformity} \\ + $B(A, \varepsilon)$ & $\varepsilon$-fattening $\{x \in X \mid d(x, A) < \varepsilon\}$ of $A$. & \autoref{definition:fattening} \\ + + % Uniform Spaces + $UC(X; Y)$ & Uniformly continuous functions $X \to Y$. & \autoref{definition:uniformcontinuity} \\ + $U^{-1}$ & Inversion of $U \subset X \times X$. & \autoref{definition:inversion} \\ + $U \circ V$ & Composition of $U, V \subset X \times X$. & \autoref{definition:composition} \\ + $U(A)$ & Slice of $U \subset X \times Y$ at $A \subset X$: $\{y \mid \exists x \in A,\, (x,y) \in U\}$. & \autoref{definition:slice} \\ + $E(S, U)$ & Entourage of the form $\{(f,g) \in X^T \mid (f(x),g(x)) \in U\ \forall x \in S\}$. & \autoref{definition:set-uniform} \\ + $\mathfrak{E}(\mathfrak{S}, \mathfrak{U})$ & $\mathfrak{S}$-uniformity, generated by $\{E(S,U) \mid S \in \mathfrak{S},\ U \in \mathfrak{U}\}$. & \autoref{definition:set-uniform} \\ + + % Function Spaces + $\mathrm{supp}(f)$ & Support of $f$. & \autoref{definition:support} \\ + $C_c(X; E)$ & Compactly supported continuous functions $X \to E$. & \autoref{definition:compactly-supported} \\ + $f \prec U$ & $f \in C_c(X; [0,1])$ with $\mathrm{supp}(f) \subset U$. & \autoref{definition:compactly-supported-01} \\ + $C_0(X; E)$ & Continuous functions vanishing at infinity. & \autoref{definition:vanish-at-infinity} \\ + $BC(X; E)$ & Bounded continuous functions $X \to E$. & \autoref{definition:bounded-continuous-function-space} \\ +\end{tabular}