Basic Topology in Euclidean Spaces

Inner Product

Def: Inner Product

If $V$ is a vector space, an inner product on $V$ is a map $⟨ \cdot, \cdot ⟩ : V \times V \to R$ such that:

Symmetry $\forall x, y \in V$ $⟨ x, y ⟩ = ⟨ y, x ⟩$
Linearity $\forall x, y, z \in V, c \in R,$ $⟨ x + c y, z ⟩ = ⟨ x, z ⟩ + c ⟨ y, z ⟩$
Positive Definite $\forall x \in V,$ $⟨ x, x ⟩ \geq 0$ , $⟨ x, x ⟩ = 0 ⟺ x = 0$

symmetry + linearity = bilinear

interesting example $⟨ f, g ⟩ = \int_{0}^{1} f (x) g (x) d x$

Cauchy-Schwarz Inequaltiy

Thm: Cauchy-Schwarz

If V is an inner product space then $\forall x, y \in$ V, $⟨ x, y ⟩^{2} \leq ⟨ x, x ⟩ ⟨ y, y ⟩$

pf: let $p = \frac{⟨ x, y ⟩}{⟨ x, x ⟩}$

$0 \leq ⟨ y - p x, y - p x ⟩ = ⟨ y, y - p x ⟩ + ⟨ - p x, y - p x ⟩ = ⟨ y, y ⟩ - 2 p ⟨ x, y ⟩ + p^{2} ⟨ x, x ⟩$

$= ⟨ y, y ⟩ - \frac{⟨ x, y ⟩^{2}}{⟨ x, x ⟩}$

$⟹ \frac{⟨ x, y ⟩^{2}}{⟨ x, x ⟩} \leq ⟨ y, y ⟩$ $⟹ ⟨ x, y ⟩^{2} \leq ⟨ y, y ⟩ ⟨ x, x ⟩$

Norm

Given a vector space V, a norm on V is a function $| | \cdot | |$ : V $\to R$ such that:

Non-degenerate $| | x | | \geq 0 \forall x \in V$ and $| | x | | = 0 ⟺ x = 0$
Homogeneity $| | α x | | = | α | | | x | |$ $\forall α \in R, x \in V$
Triangle Inequality $| | x + y | | \leq | | x | | + | | y | |, \forall x, y \in V$

Thm: Inner Product space to Normed Space

If (V, $⟨ \cdot, \cdot ⟩$ ) is an inner-product space, then defining $| | \cdot | | : V \to R$ as $| | v | | = \sqrt{⟨ v, v ⟩}$ gives a norm making (V, $| | \cdot | |$ ) into a normed vector space

*Not all normed vector spaces have an inner product space

Def: Normal

$| | v | | = 1$

Def: Euclidean Distance

Let $| | \cdot | | : R^{d} \to [0, \infty]$ be the Euclidean norm. Euclidean distance $d (u, w)$ is:

\begin{matrix} | | v - w | | \\ d ([\begin{matrix} x \\ y \\ z \end{matrix}] - [\begin{matrix} a \\ b \\ c \end{matrix}]) = \sqrt{(x - a)^{2} + (y - b)^{2} + (c - z)^{2}} \end{matrix}

Distance Inequality $d (v, z) \leq d (v, w) + d (w, z)$

Def: Real Open Ball

Let $x_{0} \in R^{d}$ . An open ball centred at $x_{0}$ with radius $r > 0$ is defined as:

B_{r} (x_{0}) = {x \in R^{d} : | | x - x_{o} | | < r}

Open and Closed Sets

Def: Interior Points of Real Subsets

Consider $S \subseteq R^{d}$ . $x \in S$ is an interior if $\exists r > 0$ such that $B_{r} (x_{o}) \subset S$

Def: Interior of a Real Set

Let $S \in R^{d}$ . The interior of S is the set of all interior points in $S$ is denoted by $S^{o}$ .

Def: Open Set

A subset $N \subseteq R^{d}$ is open if $N^{o} = N$ . All points in $N$ are interior points.

Def: Boundary points of a set

Let $S \subset R^{d}$ . A point $x \in R^{d}$ is a boundary point of $S$ if
$\forall r > 0, B_{r (x)} \cap S \neq \emptyset$ and $B_{r} (x) \cap S^{c} \neq \emptyset$ .
The set of all boundary points is denoted by: $\partial S$

Def: Closed Subset

A subset $F \subseteq R^{d}$ is called closed:

if $F^{c}$ is open
if the limit of every convergent sequence $x_{n} \in F$ is contained in $F$ .
if $\bar{F} = F$ (see: closure)

Def: Convergent Sequence

$x_{n} \in R^{d}$ converges to $ℓ \in R^{d}$ if:

\forall ε > 0, \exists N \in N such that n > N ⟹ | | x_{n} - ℓ | | < ϵ

Def: Closure of a set

Let $A \subset R^{d}$ . The closure of $A$ is denoted as $\bar{A}$ and $\bar{A} = A \cup \partial A$ .

Def: Bounded Sets

$S \subset R^{d}$ is bounded if it in contained in an open ball of finite radius

\exists x \in R^{d} and r > 0 such that S \subset B_{r} (x)

Def: Compact Set

A subset $S \subseteq R^{d}$ is compact if $S$ is closed and bounded