A metric on a topological space $\mathcal{M}$ is a mapping $d:\mathcal{M}\times\mathcal{M}\to\mathbb{R}$ such that the following three conditions hold:

1. $d(x, y) = 0 \iff x = y$ for every $x,y\in\mathcal{M}$, i.e. the distance between two points is zero if and only if they are the same,
2. $d(x, y) = d(y, x)$,
3. $d(x, z) \leq d(x, y) + d(y, z)$.

The second condition is referred to as symmetry and the third condition is referred to as the triangle inequality.

The real line $\mathbb{R}$ with the metric defined by the absolute distance between two points: $d(x, y) = |y - x|$.

The vector space $\mathbb{R}^n$ with the Euclidean distance $d_2(x, y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}$.

The space of continuous functions $\mathcal{C} = \{f:(-\epsilon, \epsilon)\to\mathbb{R}^n\}$ with the metric $d_\infty(f_1, f_2) = \mathrm{sup}_{t\in(-\epsilon, \epsilon)}|f_1(t) - f_2(t)|.$

Any Riemannian manifold is a metric space.

A Cauchy sequence is a sequence $(a_n)_{n\in\mathbb{N}}$ for which, given any $\epsilon>0$, we can find an integer $N$ such that $d(a_n, a_m) < \epsilon$ for all $n, m \geq N$.

A complete metric space is one for which every Cauchy sequence converges.

A vector space $\mathcal{V}$ is a topological space for which we define an operation called addition and denoted by $+$ and an operation called scalar multiplication (by elements of $\mathbb{R}$) denoted by $x \mapsto ax$ for $x\in\mathcal{V}$ and $x\in\mathbb{R}$ for which the following hold for all $x, y, z\in\mathcal{V}$ and $a, b\in\mathbb{R}$:

1. $x + (y + z) = (x + y) + z,$
2. $x + y = y + x,$
3. $\exists 0 \in \mathcal{V}\text{such that }x + 0 = x,$
4. $\exists -x \in \mathcal{V}\text{ such that }x + (-x) = 0,$
5. $a(ax) = (ab)x,$
6. $1x = x$ for $1\in\mathbb{R},$
7. $a(x + y) = ax + ay,$
8. $(a + b)x = ax + bx.$

The first law is known as associativity, the second one as commutativity and the last two ones are known as distributivity.