Lecture 4 - 1.2 Metric Spaces

The Inverse Image

The Inverse Image uses a Inverse Function $f^{-1}(z)$ of $Z\subset Y$ written $f:x\to y$ is $f^{-1}(z)\equiv \{x\in X\mid f(x)\in Z\}$.

Complex Numbers

In Complex Numbers in Sets, $\mathbb{C}=\mathbb{R}\times \mathbb{R}$ with usual addition of vectors

Multiplying Vectors

Multiply vectors by adding their angles multiplying their lengths.

$z=a+i\times b=(a,b)$
($b$ here can be seen as $(a,0)+(0,b)=(a+0,0+b)$)
$b=(b,0)⟹i\times b=(0,b)$

$i\times z$ rotates $z$ $90°$ counterclockwise.

Proposition

$\mathbb{C}$ is complete (cauchy) and For Complex Numbers.

Metric Spaces

Definition

Example

Discrete metric on X; $d(x,y)=\{\begin{array}{ll}0,& \text{if}x=y\\ 1,& \text{if}x\ne y\end{array}$

Vector Spaces

• Normed Vector Spaces
• Complex Vector Spaces

Example

$V=R^n=R\times \cdots \times R=\{(x_1,\dots ,x_n)|x_i\in \mathbb{R}\}$
(where the length of $R\times \cdots \times R$ has $n$ $\mathbb{R}$s.)
$V={\mathbb{R}}^2:(x,y)+(z,w)\equiv (x+z,y+w)$
$a\times (x,y)\equiv (ax,ay)$

Linear Basis Example

$u=3v_1+5v_2+i{v}_3$
$3v_1\ne 2v_2$
$c_1{v}_1+c_2{v}_2=0⟹c_1=0=c_2$
$⟹0\times v_1+0\times v_2=0$

Continuing the [[#Vector Spaces#Example]] but for Linear bases
$v_1=(1,0,0,0,\dots )$
$v_2=(0,1,0,0,\dots )$
$v_3=(0,0,1,0,\dots )$
and the way of writing this would be:
$(x_1,\dots ,x_n)={\sum }_{i=1}^n{x}_i\times v_i=0$
$⟹x_i=0$

Proposition

Any Vector Space $V$ has a Linear Basis, and every basis has the same cardinality referred to as the $dim(V)$ (dimension of $V$) of $V$.

Proof

[[ACIT4330/Lectures/Lecture 3#Axiom of Choice|Lecture 3#Axiom of Choice]]