Lowered the table of contents depth.

document.tex

@@ -4,6 +4,8 @@
 
 \begin{document}
 
+Hello this is all my notes. 
+
 \input{./src/cat/index}
 \input{./src/topology/index}
 \input{./src/fa/index}

spec.toml

@@ -18,5 +18,7 @@ neutralColour = "grey"
 searchLimit = 16
 maxSearchPages = 48
 recentChanges = 0
-tableOfContentsDepth = 2
+tableOfContentsDepth = 1
+hoverPreview = false
+copyLabelButton = false
 advertiseSpec = true

src/cat/index.tex

@@ -4,3 +4,4 @@
 \input{./cat/index.tex}
 \input{./gluing/index.tex}
 \input{./tricks/index.tex}
+\input{./notation/index.tex}

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}
+

src/dg/index.tex

@@ -2,3 +2,4 @@
 \label{part:diffgeo}
 
 \input{./derivative/index.tex}
+\input{./notation.tex}

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}

src/fa/index.tex

@@ -8,3 +8,4 @@
 \input{./rs/index.tex}
 \input{./lp/index.tex}
 \input{./order/index.tex}
+\input{./notation.tex}

src/fa/lc/hahn-banach.tex

@@ -77,7 +77,7 @@
 \begin{proof}
     By translation, assume without loss of generality that $0 \in A$. In which case, $A \in \cn^o(0)$ is convex.
 
-    Let $[\cdot]_A: E \to [0, \infty)$ be the \hyperref[gaugeg]{definition:gauge} of $A$, then $[\cdot]_A$ is a sublinear functional on $E$. For any $y, z \in E$ and $t > 0$ with $y, z \in tA$,
+    Let $[\cdot]_A: E \to [0, \infty)$ be the \hyperref[gauge]{definition:gauge} of $A$, then $[\cdot]_A$ is a sublinear functional on $E$. For any $y, z \in E$ and $t > 0$ with $y, z \in tA$,
     \[
     \abs{[y]_A - [z]_A} \le [y - z]_A \le t
     \]

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}
+

src/fa/tvs/quotient.tex

@@ -23,7 +23,7 @@
 \begin{proof}
     Let $\td E = E/M$ be the algebraic quotient of $E$ by $M$, and equip it with the quotient topology by $\pi$.
 
-    (1): By \autoref{definition:quotient-topology}, for each $\pi(U) \subset E/M$, $\pi(U)$ is open if and only if $U$ is open. Since the topology on $E$ is translation-invariant, so is the quotient topology on $E/M$. Let
+    (1): For each $U \subset E$ open, $\pi^{-1}\pi(U) = U + M$ is open, so $\pi(U)$ is open as well by \autoref{definition:quotient-topology}. Since the topology on $E$ is translation-invariant, so is the quotient topology on $E/M$. Let
     \[
     \fB = \bracs{\pi(U)| U \in \cn(0) \text{ circled and radial}}
     \]

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}

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}
+

src/topology/index.tex

@@ -5,3 +5,4 @@
 \input{./uniform/index.tex}
 \input{./functions/index.tex}
 \input{./metric/index.tex}
+\input{./notation.tex}

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}