A.1 Norms

Definition A..1 (Norm)   A normed space $(X,\Vert.\Vert)$ is a vector space $X$ (over a field $\mathbb{K}$) with a norm $\Vert.\Vert: X\to\mathbb{R}^+_0$ such that for all $x,y\in X$ and $\alpha\in\mathbb{K}$

1. $x=0 \iff \Vert x\Vert=0$,
2. $\Vert\alpha x\Vert = \vert\alpha\vert \; \Vert x\Vert$,
3. $\Vert x+y \Vert \le \Vert x\Vert + \Vert y\Vert$.

Definition A..2 (Equivalent Norms)   Two norms $\Vert.\Vert _1$ and $\Vert.\Vert _2$ on a vector space $X$ are called equivalent if there are positive numbers $a$ and $b$ such that

$$\forall x \in X: \quad a \Vert x\Vert _1 \le \Vert x\Vert _2 \le b \Vert x\Vert _1.$$

Remark. Equivalent norms on $X$ define the same topology on $X$.

Theorem A..3 (Equivalence of Norms)   All norms on a finite dimensional normed space are equivalent.