# Metric space

Canonical example: Euclidean space.

TODO examples:

## Complete metric space

In plain English: the space has no visible holes. If you start walking less and less on each step, you always converge to something that also falls in the space.
One notable example where completeness matters: Lebesgue integral of is complete but Riemann isn't.

## Inner product space

Subcase of a normed vector space, therefore also necessarily a vector space.

## Inner product

Appears to be analogous to the dot product, but also defined for infinite dimensions.

## Norm (mathematics, )

Vs metric:
• a norm is the size of one element. A metric is the distance between two elements.
• a norm is only defined on a vector space. A metric could be defined on something that is not a vector space. Most basic examples however are also vector spaces.

## Norm induced by an inner product

An inner product induces a norm with:

## Metric induced by a norm

In a vector space, a metric may be induced from a norm by using subtraction:

## Pseudometric space

Metric space but where the distance between two distinct points can be zero.
Notable example: Minkowski space.