Functions On Euclidean Space

Euclidean n-space

For

n \in ℕ

, Euclidean

n

-space is the set

ℝ^{n} = {(x^{1}, \dots, x^{n}) : x^{i} \in ℝ} .

Its elements may be treated as points or as vectors in a real vector space, with vector addition and scalar multiplication defined coordinatewise.

Euclidean Norm

x = (x^{1}, \dots, x^{n}) \in ℝ^{n}

, then the Euclidean norm of

x

| x | = \sqrt{\sum_{i = 1}^{n} {(x^{i})}^{2}} .

Euclidean Distance

x, y \in ℝ^{n}

, then the Euclidean distance from

x

y

d_{ℝ^{n}} (x, y) = | x - y |,

where

| x - y |

is the Euclidean norm of the vector

x - y

Euclidean Distance Is a Metric

The Euclidean distance

d_{ℝ^{n}} (x, y) = | x - y |

is a metric on

ℝ^{n}

Let

x, y, z \in ℝ^{n}

. By the Euclidean norm properties,

d_{ℝ^{n}} (x, y) = | x - y | \geq 0.

Also,

d_{ℝ^{n}} (x, y) = 0

if and only if

| x - y | = 0

, which holds if and only if

x - y = 0

, equivalently

x = y

For symmetry, use $x - y = - (y - x)$ and the scalar multiplication property of the Euclidean norm:

d_{ℝ^{n}} (x, y) = | x - y | = | - (y - x) | = | y - x | = d_{ℝ^{n}} (y, x) .

Finally, since

x - z = (x - y) + (y - z)

, the triangle inequality for the Euclidean norm gives

d_{ℝ^{n}} (x, z) = | x - z | = | (x - y) + (y - z) | \leq | x - y | + | y - z | = d_{ℝ^{n}} (x, y) + d_{ℝ^{n}} (y, z) .

Therefore

d_{ℝ^{n}}

satisfies the definition of a metric.

Euclidean Inner Product

x, y \in ℝ^{n}

, then their Euclidean inner product is

(x, y) = \sum_{i = 1}^{n} x^{i} y^{i} .

Euclidean Norm Properties

x, y \in ℝ^{n}

and

a \in ℝ

, then:

$| x | \geq 0$ , and $| x | = 0$ if and only if $x = 0$ .
$| \sum_{i = 1}^{n} x^{i} y^{i} | \leq | x | | y |$ , with equality if and only if $x$ and $y$ are linearly dependent.
$| x + y | \leq | x | + | y |$ .
$| a x | = | a | | x |$ .

Euclidean Inner Product Properties

x, x_{1}, x_{2}, y, y_{1}, y_{2} \in ℝ^{n}

and

a \in ℝ

, then:

$(x, y) = (y, x)$ .
$(a x, y) = (x, a y) = a (x, y)$ .
$(x_{1} + x_{2}, y) = (x_{1}, y) + (x_{2}, y)$ .
$(x, y_{1} + y_{2}) = (x, y_{1}) + (x, y_{2})$ .
$(x, x) \geq 0$ , and $(x, x) = 0$ if and only if $x = 0$ .
$| x | = \sqrt{(x, x)}$ .
$(x, y) = \frac{{| x + y |}^{2} - {| x - y |}^{2}}{4}$ .

Standard Basis

The standard basis of

ℝ^{n}

e_{1}, \dots, e_{n}

, where

e_{i}

has

1

in its

i

-th coordinate and

0

in every other coordinate.

Product of Subsets of Euclidean Space

A \subseteq ℝ^{m}

and

B \subseteq ℝ^{n}

, then their Cartesian product is

A \times B = {(x, y) \in ℝ^{m + n} : x \in A and y \in B} .

Closed Rectangle in Euclidean Space

A closed rectangle in

ℝ^{n}

is a product of closed intervals, a set of the form

[a_{1}, b_{1}] \times \dots \times [a_{n}, b_{n}] .

Open Rectangle in Euclidean Space

An open rectangle in

ℝ^{n}

is a product of open intervals, a set of the form

(a_{1}, b_{1}) \times \dots \times (a_{n}, b_{n}) .

Open Subset of Euclidean Space

A set

U \subseteq ℝ^{n}

is open if for every

x \in U

, there is an open rectangle

A

such that

x \in A \subseteq U .

Near a Point in Euclidean Space

A statement depending on points of

ℝ^{n}

holds near

a \in ℝ^{n}

if there is an open set

U \subseteq ℝ^{n}

with

a \in U

such that the statement holds for every point of

U

For example, a function has a property near $a$ if it has that property on some open set containing $a$ .

Closed Subset of Euclidean Space

A set

C \subseteq ℝ^{n}

is closed if

ℝ^{n} ∖ C

is open.

Interior, Exterior, and Boundary

Let

A \subseteq ℝ^{n}

. A point

x \in ℝ^{n}

is:

in the interior of $A$ if there is an open rectangle $B$ such that $x \in B \subseteq A$ ,
in the exterior of $A$ if there is an open rectangle $B$ such that $x \in B \subseteq ℝ^{n} ∖ A$ ,
on the boundary of $A$ if every open rectangle $B$ containing $x$ meets both $A$ and $ℝ^{n} ∖ A$ .

Open Cover

A collection

𝒰

of open subsets of

ℝ^{n}

is an open cover of

A \subseteq ℝ^{n}

if every point of

A

lies in some

U \in 𝒰

Compact Subset of Euclidean Space

A set

A \subseteq ℝ^{n}

is compact if every open cover of

A

has a finite subcover.

Heine-Borel for Closed Intervals

Every closed interval

[a, b] \subseteq ℝ

is compact.

Tube Lemma for a Compact Factor

Let

B \subseteq ℝ^{m}

be compact and let

x \in ℝ^{n}

. If

𝒰

is an open cover of

{x} \times B

, then there is an open set

V \subseteq ℝ^{n}

with

x \in V

such that

V \times B

is covered by finitely many sets from

𝒰

Product of Compact Sets is Compact

A \subseteq ℝ^{n}

and

B \subseteq ℝ^{m}

are compact, then

A \times B \subseteq ℝ^{n + m}

is compact.

Finite Product of Compact Sets is Compact

A_{1}, \dots, A_{k}

are compact, then

A_{1} \times \dots \times A_{k}

is compact. In particular, every closed rectangle in

ℝ^{k}

is compact.

Closed Bounded Subset of Euclidean Space is Compact

Every closed bounded subset of

ℝ^{n}

is compact.

Function Between Euclidean Spaces

A function

f : A \to ℝ^{m}

, where

A \subseteq ℝ^{n}

, assigns to each

x \in A

a point

f (x) \in ℝ^{m}

. The set

A

is called the domain of

f

Image and Preimage

f : A \to ℝ^{m}

B \subseteq A

, and

C \subseteq ℝ^{m}

, then the image and preimage are

f (B) = {f (x) : x \in B}

and

f^{- 1} (C) = {x \in A : f (x) \in C} .

Component Functions

A function between Euclidean spaces

f : A \to ℝ^{m}

determines component functions

f^{1}, \dots, f^{m} : A \to ℝ

f (x) = (f^{1} (x), \dots, f^{m} (x)) .

These are called the component functions of

f

Projection Function

The

i

-th projection function

π^{i} : ℝ^{n} \to ℝ

is defined by

π^{i} (x^{1}, \dots, x^{n}) = x^{i} .

Limit of a Vector-Valued Function

Let

f : A \to ℝ^{m}

, where

A \subseteq ℝ^{n}

. We write the limit

\lim_{x \to a} f (x) = b

when for every

ϵ \in ℝ^{+}

, there is a

δ \in ℝ^{+}

such that if

x \in A

and

0 < | x - a | < δ

, then

| f (x) - b | < ϵ

Continuity Between Euclidean Spaces

A function

f : A \to ℝ^{m}

, where

A \subseteq ℝ^{n}

, is continuous at

a \in A

\lim_{x \to a} f (x) = f (a) .

It is continuous if it is continuous at every point of

A

Continuity Open Set Characterization

Let

A \subseteq ℝ^{n}

. A function

f : A \to ℝ^{m}

is continuous if and only if for every open set

U \subseteq ℝ^{m}

, there is an open set

V \subseteq ℝ^{n}

such that

f^{- 1} (U) = V \cap A .

Continuous Image of Compact Set is Compact

A \subseteq ℝ^{n}

is compact and

f : A \to ℝ^{m}

is continuous, then the image

f (A) \subseteq ℝ^{m}

is compact.

Oscillation

Let

f : A \to ℝ

be bounded, where

A \subseteq ℝ^{n}

. For

a \in A

and

δ \in ℝ^{+}

, define the upper and lower local bounds

M (a, f, δ) = \sup {f (x) : x \in A and | x - a | < δ}

and

m (a, f, δ) = \inf {f (x) : x \in A and | x - a | < δ} .

The oscillation of

f

a

o (f, a) = \lim_{δ \to 0} (M (a, f, δ) - m (a, f, δ)) .

Continuity Oscillation Characterization

A bounded function

f : A \to ℝ

is continuous at

a \in A

if and only if

o (f, a) = 0.

Large Oscillation Set is Closed

Let

A \subseteq ℝ^{n}

be closed. If

f : A \to ℝ

is bounded and

ϵ \in ℝ^{+}

, then

{x \in A : o (f, x) > ϵ}

is closed.