\[V= \mathbb{R}^n\]
are closely related but are not the same thing. In particular, the axioms for a vector space do not include any mention of distance.In Euclidean space the distance between points
\[x=(x_1, x_2,...,x_n), \: y=(y_1,y_2,...,y_n)\]
is \[d=\sqrt{(x_1-y_1)^2 + ...+ (x_n-y_n)^2}\]
This distance function is also called the Euclidean norm, and satisfies all the properties of a norm, namely:
1.
\[||x|| \geq 0\]
and \[||x|| =0 \leftrightarrow x=0\]
2.
\[||kx|| =k ||f(x)|| \]
for any \[k, \: x\]
3.
\[||x+y|| \leq ||x|| + ||y||\]
We may think of a vector space as a set of n - tuples
\[\begin{pmatrix}v_1\\v_2\\.\\.\\v_n\end{pmatrix} \]
and Euclidean space as a a set of points, relative to some origin, with the distance between each point given by the distance metric.A vector space with a norm in this way is also called a normed linear space.