In geometry, the concept of Euclidean space encompasses three-dimensional space and Euclidean plane as spaces of dimensions 3 and 2 respectively.

Also Euclidean spaces generalize to higher dimensions.
For Example: euclidian space with n-tuples of real numbers, ($x_1$, $x_2$, $x_3$, ...., $x_n$) are called n-space or Cartesian space.
This theory is given by Euclid, the famous Greek Mathematician who is considered to be the father of Geometry.

Mathematically, n-dimensional Euclidian space is denoted as $\mathbb{E}$ or sometimes $\mathbb{R}$. In this section we will learn about Euclidean space in detail.

Definition

In Geometry, the key word ‘Euclidean Space’ refers either to the two dimensional plane or three dimensional space.

The notation (x, y) or (x, y, z) used to represent the coordinates of a point is extended to represent a vector ($x_1$, $x_2$, ...... $x_n$) in n-space, which is denoted by R$^n$, the letter n representing the dimension of the space.

A vector space is a  real set V, on which the  operations of vector addition and scalar multiplication are defined.  An Euclidean space is a special vector space, where the operation ‘inner product’ is additionally defined.
An Euclidean space can be defined in simple terms as follows:

An Euclidean space is a vector space V, where the operation ($v_i$, $v_j$) every pair of vectors $v_i$, $v_j$ $\varepsilon$ V such that the following conditions are satisfied.

1) ($v_i$ + $v_j$, $v_k$) = ($v_i$, $v_k$) + ($v_j$, $v_k$)  for all vectors $v_i$, $v_j$, $v_k$ $\varepsilon$  V.

2) ($\alpha v_i$, $v_j$) = $\alpha$($v_i$, $v_j$)  for all vectors $v_i$, $v_j$  $\varepsilon$ V and a is a real number.

3) ($v_i$, $v_j$) = ($v_j$, vi) for all vectors $v_i$, $v_j$ $\varepsilon$ V.

4) ($v_i$, $v_j$) > 0 and ($v_i$, $v_i$) = 0 if and only if $v_i$ = 0.
The first three properties related to any inner product and property (4) is specific to Euclidean space.

Euclidean Plane Geometry

Euclidean Plane Geometry refers to the facts of Geometry which can be illustrated on a plane surface like a piece of paper or chalk board. The study is built on Undefined terms, Definitions, Axioms , Theorems and methods of proofs.

Set, Point, line and plane are four undefined terms accepted without definitions.  These terms are explained using sketches and examples.
A definition is a statement used to explain the meaning of a term. Definitions are formed using undefined terms and previously defined terms:

An axiom or Postulate is a statement whose truth is accepted without proof.
A theorem is a statement that is proved using deductive reasoning.

Axioms of Euclidean Geometry

Below are some properties and theorems are as follows:
1)   A straight line which can be used in deductive reasoning to establish may be drawn between any two points.

2) A circle can be drawn with given radius and center or any given points where circle is passing from.

3) Any terminated straight line may be extended indefinitely.

4) All right angles of triangle are congruent.

5)    If we extended a straight line infinitely, it will be a straight line.

6)    All right angles of different triangles are same.

7)    Two coincided things are equal to one another.

8)     Euclid’s Parallel postulate, which can be equivalently stated as  “For any given point not on a given

line, there is exactly one line through the point that does not meet the given line.”

9)     Every angle is congruent to each other.

Euclidean n-Space

An ordered n-tuple of real numbers ($x_1$, $x_2$,.......,$x_n$) can be considered as a vector.

Euclidean n-space is set of all such ordered n-tuples with vector addition and scalar multiplication defined as follows:

Vector addition:  ($x_1$, $x_2$, .....,$x_n$) + ($y_1$, $y_2$, ......, $y_n$) = ($x_1$ + $y_1$, $x_2$ + $y_2$, ........ ,$x_n$ + $y_n$)

Scalar multiplication: $\alpha$($x_1$, $x_2$, ......,$x_n$) = ($\alpha x_1$ , $\alpha x_2$, ......, $\alpha x_n$)

The dot product of two vectors serves as the inner product satisfying the four properties stated above.
($x_1$, $x_2$, ..... ,$x_n$) . ($y_1$, $y_2$, ......,$y_n$) = $x_1y_1$ + $x_2 y_2$+ ........ + $x_n y_n$

Examples

Example 1: Define Euclidean  inner product of 2x2 matrices.

Solution: The set of  2 x 2 square matrices form an Euclidean space. The Matrix addition and scalar multiplication of a matrix serve as the operations of addition and scalar multiplication of the vector space.  The Euclidean inner product can be defined as follows:

Let

A = $\begin{bmatrix} a_{11} &a_{12} \\ a_{21}& a_{22} \end{bmatrix}$    and   B = $\begin{bmatrix} b_{11} &b_{12} \\ b_{21}& b_{22} \end{bmatrix}$

then

A . B = $a_{11}$ $b_{11}$  + $a_{12}$ $b_{12}$  +  $a_{21}$ $b_{21}$  + $a_{22}$ $b_{22}$

Example 2: Show that set of polynomial is an Euclidean space.

Solution: Let P be the set of polynomials with degrees not greater than n, defined in an interval [a,b].  Then P is an Euclidean space with the following operations.

(f, g) = $\int_{a}^{b}f(t)g(t)dt$