00 - Properties of ℝ

Norm on ${\mathbb{R}}^n$

For $x=(x_1,x_2,\cdots ,x_n)\in {\mathbb{R}}^n$

$$|x|=\sqrt{\sum _{i=1}^n{x}_i^2}$$

Limits in Higher Dimensions

For $f:\Omega \subseteq {\mathbb{R}}^n\to {\mathbb{R}}^m$ then

$f(\underline{x})\to \underline{b}$ as $\underline{x}\to \underline{a}$ if

$$\forall ϵ>0,\exists \delta >0\text{ such that }\forall x\in \Omega ,0<|\underline{x}-\underline{a}|<\delta ⟹|f(x)-b|<ϵ$$

Then we can write

$$\underset{\underline{x}\to \underline{a}}{\lim }f(\underline{x})=\underline{b}$$

Underlining is meant to represent vectors

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Distances in ${\mathbb{R}}^2$

For $\underline{x}=(x_1,x_2)$ and $\underline{y}=(y_1,y_2)$
Let

$$d_2(\underline{x},\underline{y}):=|\underline{x}-\underline{y}|=\sqrt{(x_1-y_1{)}^2+(x_2-y_2{)}^2}$$

Alternate Definition of Distance

$$d_{\infty }(\underline{x},\underline{y}):=\max (|x_1-y_1|,|x_2-y_2|)$$

Hence this gives the open ball of radius $r>0$ around point $\underline{a}\in {\mathbb{R}}^2$ to be

$$B_{\infty }(\underline{x},r)=\{\underline{y}\in {\mathbb{R}}^2:d_{\infty }(\underline{x},\underline{y})<r\}$$