## 0 Global Conventions
* **Character [sets](/page/Set)** — $\mathbb R$, $\mathbb C$, $\mathbb Z$, $\mathbb Q$, $\mathbb N$ (natural numbers start at **1**).
$\mathbb R^{n}_{+}:=\{x\in \mathbb R^{n}\mid x_i>0\}$; analogously $\mathbb R^{n}_{-}$.
$\mathbb R^{n}_{0}:=\mathbb R^{n}\setminus\{0\}$ (the punctured Euclidean space).
* **Points** — a generic element of an [open set](/page/Open%20Set) $U\subset \mathbb R^{n}$ is
\begin{align*}
x &= (x_1,\dots ,x_n), & x_{0} &= (x_1,\dots ,x_n).
\end{align*}
The subscript "0" singles out a reference point; it is **not** a power.
* **Index placement** —
• Superscripts $^{m}$ are *only* for powers or derivative order, never enumeration.
• Components and enumeration indices use subscripts: $v_i$, $A_{ij}$.
• Multi‑index notation $\alpha=(\alpha_1,\dots,\alpha_n)$ is encouraged: $|\alpha| = \alpha_1 + \dots + \alpha_n$, $\alpha! = \alpha_1! \cdots \alpha_n!$, $D^\alpha = \partial_{x_1}^{\alpha_1} \cdots \partial_{x_n}^{\alpha_n}$.
* **Gradient** — for a scalar function $f: U \to \mathbb{R}$, use the nabla symbol $\nabla f$; **never** $Df$. Higher [derivatives](/page/Derivative): $D^{m}f$ (superscript denotes order).
* **Total derivative** — for a map $f: U \subseteq \mathbb{R}^m \to \mathbb{R}^n$, the **total derivative** of $f$ at a point $a \in U$ is the linear map
\begin{align*}
Df_a &: \mathbb{R}^m \to \mathbb{R}^n
\end{align*}
defined by $f(a + h) = f(a) + Df_a(h) + o(|h|)$ as $h \to 0$. This is **not** a matrix — it is a linear map. The subscript $a$ denotes the point of evaluation; write $Df_a$, **not** $Df(a)$ (which could be confused with $D$ applied to $f(a)$, a constant).
* **Jacobian matrix** — the matrix representation of $Df_a$ with respect to the standard bases of $\mathbb{R}^m$ and $\mathbb{R}^n$ is the **Jacobian matrix**
\begin{align*}
Jf_a &\in \mathbb{R}^{n \times m}, \qquad (Jf_a)_{ij} = \partial_j f_i(a) = \frac{\partial f_i}{\partial x_j}(a).
\end{align*}
The derivative acts on a vector $h \in \mathbb{R}^m$ as $Df_a(h) = Jf_a\,h$ (matrix–vector product). Eigenvalues, determinants, and traces are properties of $Jf_a$ (a matrix), not of $Df_a$ (a [linear map](/page/Linear%20Map)).
* **Partial derivatives** — for a map $f:U\to V\subset \mathbb R^{m}$,
\begin{align*}
\frac{\partial f}{\partial {\tilde x}_i}:=\frac{\partial f}{\partial e_i}:=D_{e_i}f:U \to \mathbb{R}^m.
\end{align*}
* **Function declarations** — always specify *domain*, *codomain*, and *function space* (e.g. "let $u\in W^{1,p}(U;\mathbb R^{m})$").
* **Balls** — open ball $B(x_0, r) := \{x \in \mathbb R^n : |x - x_0| < r\}$. Closed ball $\overline{B}(x_0, r)$. Unit ball $B(0,1)$.
---
## 1 Analysis & PDE
| Concept | Standard Symbol | Notes |
| ---------------- | ----------------------------------- | ------------------------------------------------------ |
| Lebesgue measure | $\mathcal L^{n}$ | Integrate as $\displaystyle \int_E g\,d\mathcal L^{n}$. |
| [Weak derivative](/page/Weak%20Derivative) | $D^{\alpha}u$ | Multi‑index $\alpha$. |
| Sobolev spaces | $W^{k,p}(U)$, $H^{k}(U)=W^{k,2}(U)$ | Norm: $\|u\|_{W^{k,p}}^p = \sum_{|\alpha| \le k} \|D^\alpha u\|_{L^p}^p$. For $H^1$: $\|u\|_{H^1}^2 = \|u\|_{L^2}^2 + \|\nabla u\|_{L^2}^2$ where $\|\nabla u\|_{L^2}^2 := \sum_{i=1}^n \|\partial_i u\|_{L^2}^2$ (definition of the $L^2$ norm of a vector-valued function). On $H^1_0(\Omega)$ with $\Omega$ bounded, Poincaré gives $\|u\|_{L^2} \le C_\Omega\|\nabla u\|_{L^2}$ where $C_\Omega > 0$ depends only on the domain, so $\|\nabla u\|_{L^2}$ and $\|u\|_{H^1}$ are equivalent norms. |
| [Sobolev spaces](/page/Sobolev%20Space) with zero trace | $W^{k,p}_0(U)$, $H^k_0(U)$ | Closure of $C_c^\infty(U)$ in the $W^{k,p}$ norm. |
| Dual Sobolev space | $H^{-1}(U) = (H^1_0(U))^*$ | The dual of $H^1_0$. |
| [Hölder spaces](/page/Holder%20Space) | $C^{k,\gamma}(\bar{U})$ | $k$ derivatives, $\gamma$-Hölder [continuous](/page/Continuity) ($0 < \gamma \le 1$). |
| Laplacian | $\Delta u$ | Never use $\nabla^{2}u$. |
| Bilinear form | $B[u, v]$ or $B(u, v)$ | Associated to operator $L$; square brackets preferred. |
| Elliptic operator (divergence form) | $Lu = -\sum_{i,j} \partial_{x_i}(a_{ij}\, \partial_{x_j} u) + \sum_i b_i\, \partial_{x_i} u + c\, u$ | Coefficients $a_{ij}, b_i, c \in L^\infty(U)$; ellipticity constant $\theta$. |
| Uniform ellipticity | $\sum_{i,j} a_{ij}(x)\xi_i\xi_j \ge \theta|\xi|^2$ | For all $x \in U$, $\xi \in \mathbb R^n$, $\theta > 0$. |
| [Mollifier](/page/Standard%20Mollifier) | $\eta_\varepsilon(x) = \varepsilon^{-n}\eta(x/\varepsilon)$ | $\eta \in C_c^\infty(B(0,1))$, $\int \eta\, d\mathcal L^n = 1$. |
| [Convolution](/page/Convolution) | $(f * g)(x) = \int_{\mathbb R^n} f(x-y)\, g(y)\, d\mathcal L^n(y)$ | |
| Sobolev conjugate | $p^* = \frac{np}{n-p}$ | For $1 \le p < n$. |
### Integration Conventions
**Riemann integrals** use the differential $dx$, $dt$, etc. The integrand must always be written as a composition — never a bare function name:
\begin{align*}
\int_a^b (f \circ \gamma)(t)\, dt, \qquad \textbf{not } \int_a^b f\, dt.
\end{align*}
**Lebesgue integrals** must always specify the Lebesgue measure with its dimension and the integration variable:
\begin{align*}
\int_E (f \circ \varphi)(x)\, d\mathcal L^n(x), \qquad \textbf{not } \int_E f\, d\mu.
\end{align*}
Both the dimension superscript on $\mathcal L^n$ and the variable in $d\mathcal L^n(x)$ are mandatory. The integrand must likewise use explicit composition $f \circ \varphi$, not bare $f$.
*When referring to coordinate charts, name them explicitly, e.g. "in the chart $(\varphi,U)$"—avoid phrases like "just choose coordinates."*
---
## 2 Functional Analysis
| Concept | Standard Symbol | Notes |
|---|---|---|
| Banach space norm | $\|u\|_X$ | Always subscript the space when ambiguity is possible. |
| Bounded linear operators | $\mathcal{L}(X, Y)$ | Equipped with the operator norm $\|A\|_{\mathcal{L}(X,Y)}$. Write $\mathcal{L}(X)$ for $\mathcal{L}(X, X)$. |
| Dual space | $X^*$ | Space of bounded linear functionals. Evaluation written as $f(x)$, **never** $\langle f, x \rangle$. |
| Adjoint operator | $T^*$ | For $T \in \mathcal{L}(X, Y)$, the adjoint $T^* \in \mathcal{L}(Y^*, X^*)$. In [Hilbert spaces](/page/Hilbert%20Space), $(Tx, y)_H = (x, T^*y)_H$. |
| Inner product | $(\cdot, \cdot)_H$ | Subscript the Hilbert space $H$ when needed. Linear in the first argument. |
| $L^2$ inner product | $(f, g)_{L^2} = \int_U f\, \bar{g}\, d\mathcal L^n$ | Conjugate-linear in the second argument (complex case). |
| [Weak convergence](/page/Weak%20Convergence) | $u_k \rightharpoonup u$ | In a Banach space $X$: $f(u_k) \to f(u)$ for all $f \in X^*$. |
| Weak* convergence | $f_k \overset{*}{\rightharpoonup} f$ | In $X^*$: $f_k(x) \to f(x)$ for all $x \in X$. |
| Spectrum | $\sigma(T)$ | The set $\{\lambda \in \mathbb C : T - \lambda I \text{ not invertible}\}$. |
| Resolvent set | $\rho(T) = \mathbb C \setminus \sigma(T)$ | |
| Resolvent operator | $(T - \lambda I)^{-1}$ | Defined for $\lambda \in \rho(T)$. |
| Compact embedding | $X \subset\subset Y$ | The inclusion $X \hookrightarrow Y$ is compact. |
| Kernel / range | $\ker(T)$, $\operatorname{Range}(T)$ | |
| Orthogonal direct sum | $X = V \oplus V^\perp$ | Every $x \in X$ decomposes uniquely as $x = v + w$ with $v \in V$, $w \in V^\perp$, and $(v, w)_H = 0$. Used for closed subspaces of Hilbert spaces. |
---
## 3 [Distribution](/page/Distribution) Theory
| Concept | Standard Symbol | Notes |
|---|---|---|
| Test functions | $\mathcal{D}(\Omega) = C_c^\infty(\Omega)$ | Smooth [functions](/page/Function) with compact support in $\Omega$. |
| Distributions | $\mathcal{D}'(\Omega)$ | The [topological dual](/page/Topological%20Dual) of $\mathcal{D}(\Omega)$. |
| [Schwartz space](/page/Schwartz%20Space) | $\mathcal{S}(\mathbb R^n)$ | Rapidly decreasing smooth functions. |
| [Tempered distributions](/page/Tempered%20Distributions) | $\mathcal{S}'(\mathbb R^n)$ | The topological dual of $\mathcal{S}(\mathbb R^n)$. |
| Action of distribution on [test function](/page/Test%20Function) | $T(\phi)$ | For $T \in \mathcal{D}'(\Omega)$, $\phi \in \mathcal{D}(\Omega)$. Use functional notation $T(\phi)$, **not** angle brackets $\langle T, \phi \rangle$. |
| [Regular distribution](/page/Regular%20Distribution) | $T_f(\phi) = \int_\Omega f\, \phi\, d\mathcal L^n$ | Identifies $f \in L^1_{\mathrm{loc}}(\Omega)$ with a distribution. |
| [Distributional derivative](/page/Distributional%20Derivative) | $D^\alpha T$ defined by $D^\alpha T(\phi) = (-1)^{|\alpha|} T(D^\alpha \phi)$ | |
| Support of a distribution | $\operatorname{supp} T$ | Smallest [closed set](/page/Closed%20Set) outside which $T$ vanishes. |
| Distributional restriction | $T\big|_{\mathcal{D}'(\Omega')}$ | For $\Omega' \subseteq \Omega$ open, $T\big|_{\mathcal{D}'(\Omega')}(\phi) := T(\phi)$ for all $\phi \in C_c^\infty(\Omega')$. Well-defined because $C_c^\infty(\Omega') \subseteq C_c^\infty(\Omega)$. Never write $T\big|_{\Omega'}$ (pointwise restriction notation is meaningless for distributions). |
---
## 4 Fourier Analysis
| Concept | Standard Symbol | Notes |
|---|---|---|
| [Fourier transform](/page/Fourier%20Transform) | $\hat{f}(\xi) = \frac{1}{(2\pi)^{n/2}}\int_{\mathbb R^n} f(x)\, e^{-i\xi \cdot x}\, d\mathcal L^n(x)$ | Also written $\mathcal{F}f$. Frequency variable is $\xi$. Symmetric normalisation. |
| Inverse Fourier transform | $\check{g}(x) = \frac{1}{(2\pi)^{n/2}}\int_{\mathbb R^n} g(\xi)\, e^{i\xi \cdot x}\, d\mathcal L^n(\xi)$ | Also written $\mathcal{F}^{-1}g$. Same $(2\pi)^{-n/2}$ factor as the forward transform. |
| Plancherel identity | $\|\hat{f}\|_{L^2} = \|f\|_{L^2}$ | $\mathcal{F}$ is unitary on $L^2(\mathbb{R}^n)$. No extra constants. |
| Convolution formula | $\widehat{f * g} = (2\pi)^{n/2}\,\hat{f}\,\hat{g}$ | The factor $(2\pi)^{n/2}$ appears because the normalisation is split symmetrically. |
| Fourier multiplier | $\widehat{Tf}(\xi) = m(\xi)\hat{f}(\xi)$ | The symbol $m(\xi)$ is the multiplier. |
| Seminorms on $\mathcal{S}$ | $\|f\|_{\alpha,\beta} = \sup_{x \in \mathbb R^n} |x^\alpha D^\beta f(x)|$ | Indexed by multi-indices $\alpha, \beta$. |
---
## 5 Complex Analysis
| Concept | Standard Symbol | Notes |
|---|---|---|
| Complex variable | $z = x + iy$ | Real and imaginary parts: $\operatorname{Re}(z) = x$, $\operatorname{Im}(z) = y$. |
| Holomorphic function | $f: \Omega \to \mathbb C$ holomorphic | Sometimes written $f \in \mathcal{O}(\Omega)$. |
| Wirtinger derivatives | $\partial_z = \frac{1}{2}(\partial_x - i\partial_y)$, $\partial_{\bar{z}} = \frac{1}{2}(\partial_x + i\partial_y)$ | |
| Contour integral | $\oint_\gamma f(z)\, dz$ | Closed contour $\gamma$; specify orientation. |
| Residue | $\operatorname{Res}(f, z_0)$ | Coefficient of $(z - z_0)^{-1}$ in the Laurent expansion. |
| Winding number | $n(\gamma, z_0) = \frac{1}{2\pi i}\oint_\gamma \frac{dz}{z - z_0}$ | |
---
## 6 Geometric Measure Theory (GMT)
We adopt **GMT notation** for integration and measures:
* **[Hausdorff measure](/page/Hausdorff%20Measure)** — $\mathcal H^{k}$. Surface integrals:
\begin{align*}
\int_{\Sigma} g(x)\,d\mathcal H^{n-1}(x).
\end{align*}
* **Perimeter / BV** — perimeter of a set $E$ in $U$: $P(E;U)$.
Total variation of $u$: $|Du|(U)$.
* **Rectifiable sets** — denoted by $\mathcal R^{k}$.
* **Currents** — boldface **T**; [boundary](/page/Boundary) $\partial\mathbf T$.
---
## 7 Linear Algebra & Tensor Calculus
* **Vectors** — lower‑case bold **v**, **w** (optional). Components $v_i$.
* **Matrices / tensors** — uppercase or calligraphic: $A_{ij}$, $\mathcal T_{ijk}$.
* **Inner product** — $\langle v,w \rangle$ or $v\cdot w$ (Euclidean). Norm $|v|$.
* **Determinant** — $\det A$.
* **Transpose** — $A^\top$ (never $A^T$, which could be confused with an operator).
* **Identity matrix** — $I$ or $I_n$.
* **Eigenvalues** — $\lambda_1 \le \lambda_2 \le \dots$ (ordered with multiplicity for symmetric/[self-adjoint operators](/page/Self-Adjoint%20Operators)).
* **Outer product** — For $\xi, \eta \in \mathbb{R}^n$, write $\xi \otimes \eta$ for the $n \times n$ matrix with entries $(\xi \otimes \eta)_{ij} = \xi_i \eta_j$, i.e., $\xi\eta^\top$ in column-vector notation. In particular, $\xi \otimes \xi$ is the rank-one orthogonal projection onto $\operatorname{span}(\xi)$ after normalisation: $\frac{\xi \otimes \xi}{|\xi|^2}$.
---
## 8 Topology & [Metric Spaces](/page/Metric%20Space)
| Concept | Standard Symbol | Notes |
|---|---|---|
| Open ball | $B(x_0, r)$ | In a metric space $(X, d)$. |
| Closed ball | $\overline{B}(x_0, r)$ | |
| Closure | $\overline{A}$ | |
| Interior | $A^\circ$ or $\operatorname{int}(A)$ | |
| Boundary | $\partial A$ | |
| Distance to a set | $\operatorname{dist}(x, A) = \inf_{a \in A} d(x, a)$ | |
| Diameter | $\operatorname{diam}(A) = \sup_{x,y \in A} d(x,y)$ | |
| Compact embedding | $X \subset\subset Y$ | Same symbol as in functional analysis. |
| [Topological](/page/Topology) space | $(X, \tau)$ | $\tau$ is the topology. |
---
## 9 Differential Geometry
* **No superscript components.** Even on manifolds, write a vector field with subscripts:
\begin{align*}
X &= \sum_{i=1}^{n} X_i\,\partial_{x_i}.
\end{align*}
* **Coordinate charts.** Always state the chart explicitly, e.g. "Let $(U,\varphi)$ be a chart with coordinates $(x_1,\dots,x_n)$."
* **Differential of a map.** For a smooth $F:M\to N$,
\begin{align*}
dF_p &: T_pM \longrightarrow T_{F(p)}N, & (dF_p)_{ij}&=\partial_{x_j}F_i(p).
\end{align*}
* **Gradient on a Riemannian manifold** — denote by $\nabla f$; its components are
\begin{align*}
(\nabla f)_i = \sum_{j=1}^{n} g_{ij}\,\partial_{x_j}f.
\end{align*}
---
## 10 Dynamical Systems & ODEs
| Concept | Standard Symbol | Notes |
|---|---|---|
| ODE system | $\dot{x} = f(x)$ or $\frac{dx}{dt} = f(x)$ | Autonomous; state $x \in \mathbb R^n$. |
| Flow map | $\varphi_t(x_0)$ or $\varphi(t, x_0)$ | Solution at time $t$ starting from $x_0$. |
| Equilibrium / fixed point | $x^*$ with $f(x^*) = 0$ | |
| Jacobian at equilibrium | $Jf_{x^*} \in \mathbb{R}^{n \times n}$ | Matrix representation of $Df_{x^*}$; entries $(Jf_{x^*})_{ij} = \partial_j f_i(x^*)$. Eigenvalues, trace, determinant are properties of this matrix. |
| Discrete-time map | $x_{k+1} = g(x_k)$ | Iteration of $g: \mathbb R^n \to \mathbb R^n$. |
| Bifurcation parameter | $\mu$ or $\lambda$ | State clearly which symbol is the parameter. |
| Stable/unstable manifolds | $W^s(x^*)$, $W^u(x^*)$ | |
---
## 11 [Calculus of Variations](/page/Calculus%20of%20Variations)
| Concept | Standard Symbol | Notes |
|---|---|---|
| Functional | $I[u] = \int_U L(x, u, \nabla u)\, d\mathcal L^n$ | Square brackets for functionals; parentheses for functions. |
| Lagrangian | $L(x, u, p)$ | Arguments: position $x$, function value $u$, gradient $p = \nabla u$. |
| First variation | $\delta I[u; v] = \frac{d}{d\varepsilon}\Big|_{\varepsilon=0} I[u + \varepsilon v]$ | Direction $v$; also written $\delta I(u)(v)$. |
| Euler–Lagrange equation | $-\sum_{i=1}^n \partial_{x_i}\left(\frac{\partial L}{\partial p_i}\right) + \frac{\partial L}{\partial u} = 0$ | Weak form involves $B[u, v]$. |
| Admissible set | $\mathcal{A}$ | Typically a subset of a Sobolev space. |
---
## 12 Probability & Measure Theory
* **Probability space** — $(\Omega, \mathcal F, \mathbb P)$. Measure space: $(E, \mathcal E, \mu)$.
* **Sigma‑algebra** — $\mathcal F$, $\mathcal E$, $\mathcal G$. Generated $\sigma$-algebra: $\sigma(\mathcal A)$. Borel $\sigma$-algebra: $\mathcal B(\mathbb R^n)$.
* **Filtration** — $(\mathcal F_t)_{t\ge 0}$.
* **Random variables** — capital letters $X, Y$. Expectation $\mathbb E[X]$. Variance $\operatorname{Var}(X)$. Covariance $\operatorname{Cov}(X, Y)$.
* **Conditional expectation** — $\mathbb E[X \mid \mathcal G]$ (conditional on sub-$\sigma$-algebra $\mathcal G$).
* **Distribution / law** — $\mu_X = \mathbb P \circ X^{-1}$ (image measure). Distribution function: $F_X(x) = \mathbb P(X \le x)$.
* **Characteristic function** — $\phi_X(u) = \mathbb E[e^{iu \cdot X}]$.
* **Convergence in distribution** — $X_n \xrightarrow{d} X$.
* **Convergence in probability** — $X_n \xrightarrow{\mathbb P} X$.
* **Almost sure convergence** — $X_n \xrightarrow{a.s.} X$.
* **$L^p$ convergence** — $X_n \xrightarrow{L^p} X$, meaning $\|X_n - X\|_p \to 0$.
* **Absolute continuity** — $\nu \ll \mu$ means $\mu(A) = 0 \implies \nu(A) = 0$. Radon–Nikodym derivative: $d\nu/d\mu$.
* **Mutual singularity** — $\nu \perp \mu$.
* **Product measure** — $\mu_1 \otimes \mu_2$. Product $\sigma$-algebra: $\mathcal E_1 \otimes \mathcal E_2$.
* **Brownian motion** — $W_t$ or $B_t$ (state which at first use). Itô integral: $\int_0^t f_s\, dW_s$.
* **Stochastic differential** — $dX_t = b(X_t)\, dt + \sigma(X_t)\, dW_t$.
When PDE pages invoke stochastic tools, reuse these symbols to avoid clashes.
---
## 13 Set-Theoretic Notation
### 13.1 Basic Set Operations
| Symbol | Meaning | Notes |
|---|---|---|
| $\varnothing$ | empty set | Preferred over $\emptyset$. |
| $\subset$ | subset (not necessarily proper) | |
| $\subsetneq$ | proper subset | |
| $\supset$, $\supseteq$ | superset, superset or equal | |
| $A \cup B$ | union | |
| $A \cap B$ | intersection | |
| $A \setminus B$ | set difference | $\{x \in A : x \notin B\}$. |
| $A^c$ or $A^\complement$ | complement | Relative to an ambient space; specify if ambiguous. |
| $A \triangle B$ | symmetric difference | $(A \setminus B) \cup (B \setminus A)$. |
| $A \times B$ | Cartesian product | |
| $\bigcup_{i \in I} A_i$ | indexed union | |
| $\bigcap_{i \in I} A_i$ | indexed intersection | |
### 13.2 Set-Theoretic [Limits](/page/Limit)
| Symbol | Definition | Meaning |
|---|---|---|
| $\limsup_n A_n$ | $\bigcap_{n=1}^\infty \bigcup_{m \ge n} A_m$ | $\{\omega : \omega \in A_n \text{ for infinitely many } n\}$ ("$A_n$ i.o.") |
| $\liminf_n A_n$ | $\bigcup_{n=1}^\infty \bigcap_{m \ge n} A_m$ | $\{\omega : \omega \in A_n \text{ for all sufficiently large } n\}$ |
### 13.3 Functions and Maps
| Symbol | Meaning | Notes |
|---|---|---|
| $f: A \to B$ | function from $A$ to $B$ | Always specify domain and codomain. |
| $f|_A$ | restriction of $f$ to $A$ | |
| $f^{-1}(B)$ | preimage | $\{x \in A : f(x) \in B\}$. |
| $f \circ g$ | composition | $(f \circ g)(x) = f(g(x))$. |
| $\operatorname{id}_X$ or $\operatorname{Id}$ | identity map on $X$ | |
| $f^+$, $f^-$ | positive and negative parts | $f^+ = \max\{f, 0\}$, $f^- = \max\{-f, 0\}$; $f = f^+ - f^-$, $|f| = f^+ + f^-$. |
### 13.4 Indicator Functions
| Symbol | Meaning | Notes |
|---|---|---|
| $\mathbf{1}_A$ or $\mathbb{1}_A$ | indicator function of set $A$ | $\mathbf{1}_A(x) = 1$ if $x \in A$, $0$ otherwise. Both notations are acceptable; be consistent within a single page. |
| $\chi_E$ | indicator of $E$ (alternative) | Older convention; $\mathbf{1}_E$ preferred for new pages. |
### 13.5 Cardinality and Indexing
| Symbol | Meaning |
|---|---|
| $|A|$ or $\#A$ | cardinality of a finite set $A$ |
| $\operatorname{card}(A)$ | cardinality (general) |
| $\aleph_0$ | cardinality of $\mathbb N$ (countably infinite) |
### 13.6 Miscellaneous
| Symbol | Meaning |
|---|---|
| $\operatorname{supp} f$ | closed support of $f$ |
| $\operatorname{dist}(x, A)$ | distance from $x$ to set $A$ |
| $\operatorname{diam}(A)$ | diameter of $A$ |
| $\operatorname{sgn}(x)$ | sign function |
| $\lfloor x \rfloor$ | floor (greatest integer $\le x$) |
| $\lceil x \rceil$ | ceiling (least integer $\ge x$) |
| $\min$, $\max$ | minimum, maximum (when attained) |
| $\inf$, $\sup$ | infimum, supremum |
---
## 14 Function Spaces
| Space | Definition | Notes |
|---|---|---|
| $C(U)$ | continuous functions on $U$ | |
| $C^k(U)$ | $k$-times continuously differentiable | |
| $C^k(\bar{U})$ | $C^k$ functions with derivatives extending continuously to $\bar{U}$ | |
| $C^\infty(U)$ | smooth (infinitely differentiable) | |
| $C_c^\infty(\Omega)$ | smooth with compact support in $\Omega$ | Same as $\mathcal{D}(\Omega)$; see §3. |
| $C_c(U)$ | continuous with compact support | |
| $C_0(X)$ | continuous functions vanishing at infinity | $\{f \in C(X) : \text{for all } \varepsilon > 0, \{|f| \ge \varepsilon\} \text{ is compact}\}$. |
| $C_b(X)$ | bounded continuous functions | Normed by $\|f\|_\infty = \sup_{x \in X} |f(x)|$. |
| $C^{k,\gamma}(\bar{U})$ | Hölder space | See §1. |
| $L^p(E, \mathcal E, \mu)$ | $p$-[integrable](/page/Integral) functions | $\|f\|_p = (\int |f|^p\, d\mu)^{1/p}$; elements identified up to a.e. equality. |
| $L^\infty(E, \mathcal E, \mu)$ | essentially bounded functions | $\|f\|_\infty = \operatorname{ess\,sup} |f|$. |
| $L^p_{\mathrm{loc}}(\Omega)$ | locally $p$-integrable | $f \in L^p(K)$ for every compact $K \subset \Omega$. |
| $\ell^p$ | $p$-summable [sequences](/page/Sequence) | $\|(a_n)\|_{\ell^p} = (\sum_n |a_n|^p)^{1/p}$. |
When the measure space is clear from context, write $L^p(U)$ rather than $L^p(U, \mathcal B(U), \mathcal L^n)$.
---
## 15 Convergence & Embedding Arrows
| Symbol | Meaning | Context |
|---|---|---|
| $\to$ | strong (norm) convergence | General. |
| $\rightharpoonup$ | weak convergence | [Banach spaces](/page/Banach%20Space); see §2. |
| $\overset{*}{\rightharpoonup}$ | weak* convergence | Dual spaces; see §2. |
| $\xrightarrow{d}$ | convergence in distribution | Probability; see §12. |
| $\xrightarrow{\mathbb P}$ | convergence in probability | Probability; see §12. |
| $\xrightarrow{a.s.}$ | almost sure convergence | Probability; see §12. |
| $\xrightarrow{L^p}$ | convergence in $L^p$ norm | $\|f_n - f\|_p \to 0$. |
| $f_n \uparrow f$ or $f_n \nearrow f$ | monotone increase to $f$ | $f_n \le f_{n+1}$ pointwise, $\lim f_n = f$. |
| $f_n \downarrow f$ or $f_n \searrow f$ | monotone decrease to $f$ | $f_n \ge f_{n+1}$ pointwise, $\lim f_n = f$. |
| $\hookrightarrow$ | continuous embedding | $X \hookrightarrow Y$: there exists a bounded injective linear map $j: X \to Y$ with $\|j(u)\|_Y \le C\|u\|_X$. When the elements of $X$ and $Y$ are the same type of object, $j$ is the inclusion; when they differ (e.g. functions vs distributions), $j$ is the canonical embedding (such as $f \mapsto T_f$). |
| $\hookrightarrow\hookrightarrow$ | compact embedding | $X \hookrightarrow\hookrightarrow Y$: the embedding $j: X \to Y$ is compact — every bounded sequence in $X$ has a subsequence convergent in $Y$. Strictly stronger than $\hookrightarrow$. Also written $\subset\subset$ in some references. |
---
## 16 Limits, Suprema & Asymptotic Notation
### 16.1 Limits and Extrema
| Symbol | Meaning |
|---|---|
| $\lim_{n \to \infty} a_n$ | limit of a sequence |
| $\limsup_{n \to \infty} a_n$ | $\inf_n \sup_{m \ge n} a_m$ — eventual upper bound |
| $\liminf_{n \to \infty} a_n$ | $\sup_n \inf_{m \ge n} a_m$ — eventual lower bound |
| $\sup_{x \in A} f(x)$ | supremum (least upper bound) |
| $\inf_{x \in A} f(x)$ | infimum (greatest lower bound) |
| $\operatorname{ess\,sup} f$ | essential supremum | $\inf\{\lambda : f \le \lambda \text{ a.e.}\}$. |
### 16.2 Asymptotic Notation
| Symbol | Meaning | Precise definition |
|---|---|---|
| $f = O(g)$ as $x \to x_0$ | $f$ is bounded by $g$ near $x_0$ | There exist $C > 0$ and a neighbourhood $U$ of $x_0$ such that $|f(x)| \le C|g(x)|$ for all $x \in U$. This is a **local** statement about behaviour near $x_0$. |
| $f = o(g)$ as $x \to x_0$ | $f$ is negligible compared to $g$ near $x_0$ | $f(x)/g(x) \to 0$ as $x \to x_0$. |
| $f \sim g$ as $x \to x_0$ | $f$ is asymptotic to $g$ near $x_0$ | $f(x)/g(x) \to 1$ as $x \to x_0$. |
| $f \lesssim g$ | $f$ is **globally** bounded by $g$ up to a constant | There exists $C > 0$ such that $|f(x)| \le C|g(x)|$ for **all** $x$ in the domain. No limiting process or neighbourhood is involved — this is a **uniform** bound. The implied constant may depend on fixed parameters (dimension, domain, Sobolev index, etc.) but not on the functions themselves. **Not the same as $f = O(g)$**, which is local. |
| $f \gtrsim g$ | $g \lesssim f$ | |
| $f \asymp g$ | $f \lesssim g$ and $g \lesssim f$ | Equivalence up to constants: $c|g(x)| \le |f(x)| \le C|g(x)|$ for all $x$. |
When using $\lesssim$, state explicitly which parameters the implied constant depends on, e.g. "where the implicit constant depends on $n$ and $p$."
---
## 17 Logic, Quantifiers & Standard Abbreviations
### 17.1 Logical Symbols
| Symbol | Meaning |
|---|---|
| $\forall$ | for all |
| $\exists$ | there exists |
| $\exists!$ | there exists a unique |
| $\implies$ | implies |
| $\iff$ | if and only if |
| $:=$ or $\coloneqq$ | defined to be equal to | LHS is being defined. |
| $=:$ | defined to be equal to | RHS is being defined. |
| $\neg$ | logical negation |
### 17.2 Standard Abbreviations
| Abbreviation | Meaning | Usage |
|---|---|---|
| a.e. | almost everywhere | With respect to a specified measure: "$f = 0$ $\mu$-a.e." |
| a.s. | almost surely | In probability: "$X_n \to X$ a.s." |
| i.i.d. | independent and identically distributed | |
| i.o. | infinitely often | $\{A_n \text{ i.o.}\} = \limsup_n A_n$. |
| iff | if and only if | In prose only; use $\iff$ in displayed formulas. |
| LHS / RHS | left-hand side / right-hand side | |
| WLOG | without loss of generality | |
| w.r.t. | with respect to | |
### 17.3 Conjugate Exponents
The notation $p'$ or $q$ for the Hölder conjugate of $p$ is used throughout. The convention is:
\begin{align*}
\frac{1}{p} + \frac{1}{p'} = 1, \quad 1 \le p \le \infty,
\end{align*}
with $1' = \infty$ and $\infty' = 1$. When two exponents are in play, write $(p, p')$ or $(p, q)$ with $1/p + 1/q = 1$ and state the relationship explicitly.
---
## 18 [Numerical Analysis](/page/Numerical%20Analysis)
| Concept | Standard Symbol | Notes |
|---|---|---|
| Grid spacing | $h$ or $\Delta x$ | |
| Time step | $k$ or $\Delta t$ | |
| Grid points | $x_j = jh$, $t_n = nk$ | Subscripts for spatial index, superscripts for time level only when clearly distinguished from powers. |
| Numerical approximation | $U_j^n \approx u(x_j, t_n)$ | |
| Truncation error | $\tau$ | |
| Condition number | $\kappa(A)$ | $\kappa(A) = \|A\|\|A^{-1}\|$. |
---
## 19 Checklist Before Publishing ✅
* [ ] Every new symbol defined at first use.
* [ ] Power/superscript vs. component/subscript discipline respected.
* [ ] Integrals carry explicit measure unless obvious.
* [ ] Function declarations include domain, codomain, and ambient space.
* [ ] Dual pairing uses $f(x)$, never angle brackets.
* [ ] Implied constants in $\lesssim$ have their dependencies stated.
* [ ] No banned wording (see Notation §6 in Authoring Guide).
Keep this guide open while writing; consistency saves your future readers!
