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0 Global Conventions
- Character sets — $\mathbb R$, $\mathbb C$, $\mathbb Z$, $\mathbb Q$, $\mathbb N$ (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 $U\subset \mathbb R^{n}$ 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: $D^{m}f$ (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).
- 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 |
$D^{\alpha}u$ |
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 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 |
$C^{k,\gamma}(\bar{U})$ |
$k$ derivatives, $\gamma$-Hölder continuous ($0 < \gamma \le 1$). |
| 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 |
$\eta_\varepsilon(x) = \varepsilon^{-n}\eta(x/\varepsilon)$ |
$\eta \in C_c^\infty(B(0,1))$, $\int \eta\, d\mathcal L^n = 1$. |
| 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, $(Tx, y)_H = (x, T^*y)_H$. |
| 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 |
$u_k \rightharpoonup u$ |
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. |
| Concept |
Standard Symbol |
Notes |
| Test functions |
$\mathcal{D}(\Omega) = C_c^\infty(\Omega)$ |
Smooth functions with compact support in $\Omega$. |
| Distributions |
$\mathcal{D}'(\Omega)$ |
The topological dual of $\mathcal{D}(\Omega)$. |
| Schwartz space |
$\mathcal{S}(\mathbb R^n)$ |
Rapidly decreasing smooth functions. |
| Tempered distributions |
$\mathcal{S}'(\mathbb R^n)$ |
The topological dual of $\mathcal{S}(\mathbb R^n)$. |
| Action of distribution on test function |
$T(\phi)$ |
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 |
$T_f(\phi) = \int_\Omega f\, \phi\, d\mathcal L^n$ |
Identifies $f \in L^1_{\mathrm{loc}}(\Omega)$ with a distribution. |
| Distributional derivative |
$D^\alpha T$ defined by $D^\alpha T(\phi) = (-1)^{|\alpha|} T(D^\alpha \phi)$ |
|
| Support of a distribution |
$\operatorname{supp} T$ |
Smallest closed set 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 |
$\hat{f}(\xi) = \frac{1}{(2\pi)^{n/2}}\int_{\mathbb R^n} f(x)\, e^{-i\xi \cdot x}\, d\mathcal L^n(x)$ |
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 — $\mathcal H^{k}$. 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 $\partial\mathbf T$.
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).
- 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: $\frac{\xi \otimes \xi}{|\xi|^2}$.
| 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 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 |
$W^s(x^*)$, $W^u(x^*)$ |
|
| 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
| 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 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 |
$\|(a_n)\|_{\ell^p} = (\sum_n |a_n|^p)^{1/p}$. |
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; 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 |
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 |
| $=:$ |
defined to be equal to |
| $\neg$ |
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.
| 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 ✅
Keep this guide open while writing; consistency saves your future readers!
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