## 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 $B(0,1)$.

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## 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."*

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## 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. |

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## 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). |

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## 4 Fourier Analysis