Let $\Omega\subset\mathbb C$ be open, let $f:\Omega\to\mathbb C$ be holomorphic, and let $z_0\in\Omega$ satisfy $f'(z_0)\ne0$. Then there exist open neighbourhoods $U\subset\Omega$ of $z_0$ and $V\subset\mathbb C$ of $f(z_0)$ such that $f|_U:U\to V$ is bijective, $(f|_U)^{-1}:V\to U$ is holomorphic, and \begin{align*} \left((f|_U)^{-1}\right)'(w)=\frac{1}{f'((f|_U)^{-1}(w))} \end{align*} for every $w\in V$.

Let $U\subsetneq\mathbb R^n$ be open and let $L$ be a second-order nondivergence form operator with real symmetric coefficient matrix $A=(a_{ij})$. Then $A$ is uniformly elliptic with constant $\theta>0$ iff \begin{align*} p_L(x,\xi)&\ge \theta |\xi|^2 \end{align*} for $\mathcal L^n$-a.e. $x\in U$ and every $\xi\in\mathbb R^n$.

Let $U\subsetneq\mathbb R^n$ be open, let $0<\gamma<1$, and let \begin{align*} Lu&=-\sum_{i,j=1}^n a_{ij}\partial_{x_i}\partial_{x_j}u+\sum_{i=1}^n b_i\partial_{x_i}u+c u \end{align*} be uniformly elliptic in nondivergence form with ellipticity constant $\theta>0$. Suppose $a_{ij},b_i,c\in C^{0,\gamma}(U)$ for $1\le i,j\le n$. If $u\in C^{2,\gamma}(U)$ and $Lu=f$ with $f\in C^{0,\gamma}(U)$, then whenever $\overline W\subsetneq V$ and $\overline V$ is compactly contained in $U$, \begin{align*} \|u\|_{C^{2,\gamma}(\overline W)}&\le C\left(\|f\|_{C^{0,\gamma}(\overline V)}+\|u\|_{C^0(\overline V)}\right), \end{align*} where $C$ depends on $n$, $\gamma$, $\theta$, $W$, $V$, and the $C^{0,\gamma}(\overline V)$ norms of the coefficients.

Let $U\subsetneq\mathbb R^n$ be bounded and open. Suppose $a_{ij},b_i,c\in L^\infty(U)$, and let $B:H_0^1(U)\times H_0^1(U)\to\mathbb R$ be the [bilinear form](/page/Bilinear%20Form) associated to the divergence form operator \begin{align*} Lu&=-\sum_{i,j=1}^n\partial_{x_i}(a_{ij}\partial_{x_j}u)+\sum_{i=1}^n b_i\partial_{x_i}u+c u. \end{align*} Then there exists $C>0$ such that \begin{align*} |B[u,v]|&\le C\|u\|_{H^1(U)}\|v\|_{H^1(U)} \end{align*} for every $u,v\in H_0^1(U)$.

Let $U\subsetneq\mathbb R^n$ be bounded and open. Let $B:H_0^1(U)\times H_0^1(U)\to\mathbb R$ be bounded and coercive. For every $F\in H^{-1}(U)$, there exists a unique $u\in H_0^1(U)$ such that \begin{align*} B[u,v]&=F(v) \end{align*} for every $v\in H_0^1(U)$.

Let $(E_i,\mathcal E_i,\mu_i)$ be probability spaces for $i\in\{1,\dots,n\}$, and set \begin{align*} (E,\mathcal E,\mu)=(E_1\times\cdots\times E_n,\mathcal E_1\otimes\cdots\otimes\mathcal E_n,\mu_1\otimes\cdots\otimes\mu_n). \end{align*} Let $\pi_i:E\to E_i$ be the coordinate projection $\pi_i(x_1,\dots,x_n)=x_i$. Then $\pi_1,\dots,\pi_n$ are independent and $\mu_{\pi_i}=\mu_i$.

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space. If $A_1,\dots,A_n\in\mathcal F$ are mutually independent, then for every choice $B_i\in\{A_i,A_i^c\}$, the events $B_1,\dots,B_n$ are mutually independent.

Let $X_1,\dots,X_n$ be independent real-valued random variables. Then, for every $u\in\mathbb R$, \begin{align*} \phi_{X_1+\cdots+X_n}(u)=\prod_{i=1}^n\phi_{X_i}(u). \end{align*}

Let $(X_i)_{i\in I}$ be an independent family of random variables on $(\Omega,\mathcal F,\mathbb P)$. Let $I_1,\dots,I_m\subset I$ be pairwise disjoint, and set \begin{align*} \mathcal G_r=\sigma(X_i:i\in I_r), \qquad r\in\{1,\dots,m\}. \end{align*} Then $\mathcal G_1,\dots,\mathcal G_m$ are independent sigma-algebras.

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space with filtration $(\mathcal F_t)_{t\in T}$, and let $(X_t)_{t\in T}$ be an integrable adapted process. Then $(X_t)_{t\in T}$ is a martingale if and only if for every $s\le t$, \begin{align*} \mathbb E[X_t-X_s\mid\mathcal F_s]=0 \end{align*} $\mathbb P$-a.s.