${\displaystyle {}v,w\in V}$

${\displaystyle {}d{\left(v,w\right)}:=\Vert {v-w}\Vert$ :={\sqrt {\left\langle v-w,v-w\right\rangle }}\,}

den Abstand zwischen ${\displaystyle {}v}$ und ${\displaystyle {}w}$.

${\displaystyle {}M}$

${\displaystyle {}x,y\in M}$

${\displaystyle \gamma \colon [a,b]\longrightarrow X}$

mit ${\displaystyle {}\gamma (a)=x}$ und ${\displaystyle {}\gamma (b)=y}$ gibt.

${\displaystyle {}f}$

${\displaystyle {}t\in I}$

${\displaystyle \operatorname {lim} _{h\rightarrow 0}\,{\frac {f(t+h)-f(t)}{h}}}$

existiert.

${\displaystyle {}P}$

${\displaystyle {}\operatorname {rang} \,\left(D\varphi \right)_{P}={\min {\left(\dim _{}{\left(V\right)},\dim _{}{\left(W\right)}\right)}}\,}$

ist.

${\displaystyle {}D_{i}}$

${\displaystyle {}i}$

${\displaystyle {}P\in \mathbb {R} ^{n}}$

${\displaystyle {\begin{pmatrix}D_{1}D_{1}f(P)&\cdots &D_{1}D_{n}f(P)\\\vdots &\ddots &\vdots \\D_{n}D_{1}f(P)&\cdots &D_{n}D_{n}f(P)\end{pmatrix}}}$

die Hesse-Matrix zu ${\displaystyle {}f}$ im Punkt ${\displaystyle {}P}$.

${\displaystyle f\colon T\longrightarrow M}$

derart gibt, dass es zu jedem ${\displaystyle {}\epsilon >0}$ ein ${\displaystyle {}n_{0}}$ gibt mit

${\displaystyle d{\left(f_{n}(x),f(x)\right)}\leq \epsilon {\text{ für alle }}n\geq n_{0}{\text{ und alle }}x\in T.}$