# Quick Primer on Metric Spaces

**Table of Contents**

### Vector Space

A vector space is set of mathemetical objects that can be multiplied and added together to produce objects of the same kind. This notion of vector spaces proves to be a very useful framework for extending methods and structures to very different types of problems. A few special types of vector spaces you may be already be familiar with:

##### Function Spaces

We can add functions together and scale them as well.

In Figure 1 we can see there exists:

- A function in blue \(f(x) = x\)
- A function in red \(g(x) = \sin(x)\)

Taking the summation of these two functions produces a third function in green \((f + g)(x) = \sin(x) + x \)

##### Euclidean Spaces

We can add \(i, j, k\) together and scale them as well to produce a vector \(a\) .

Typically this is the vector space you learn in linear algebra because it is intuitive to understand and very helpful for extending additional structure into spaces.

### More Formally

**Definition**: A *field* is a set
\(\mathbb{F}\)
of numbers, such that
\(a, b \in \mathbb{F} \implies a+ b, a - b, ab, \frac{a}{b} \in \mathbb{F}.\)
Some examples of sets that would count as fields are the real numbers
\(\mathbb{R}\)
and the complex numbers
\(\mathbb{C}\)
.

**Definition**: A *Vector Space* consists of a set
\(\textbf{V}\)
, a field
\(\mathbb{F}\)
and two operations:

- Vector Addition: \(\textbf{a}, \textbf{b} \in \textbf{V} \implies \textbf{a} + \textbf{b} \in \textbf{V}\)
- Scalar Multiplication: \(c \in \mathbb{F}, \textbf{v} \in \textbf{V} \implies c\textbf{v} \in \textbf{V}\)

### Metric Space

Essentially a metric is a measurement to see how similar two vectors are to one another. In geometrical terms, this similarity could be interpreted as distance.

**Definition**: A *Metric Space* consists of a set
\(\textbf{V}\)
, and has a notion of distance
\(d(x,y) \ \forall x, y \in \textbf{V}\)
. That is
\(d : \textbf{V} \times \textbf{V} \to [0;\infty)\)
and satisfies these following properties
\(\forall x,y,z \in \textbf{V}\)
:

- Identification: \(d(x,y) = 0 \implies x = y\)
- Symmetry: \(d(x,y) = d(y,x)\)
- Triangle Inequality: \(d(x,y) \leq d(x,z) + d(z,y)\)

As long as these axioms hold true for a particular vector space with a function d, then it can be categorized as a metric space. A few examples you may be familiar with are the euclidean distance metric, or L2.