Metric space

Metric space vs normed vector space vs inner product space

TODO examples:

metric space that is not a normed vector space
norm vs metric: a norm gives size of one element. A metric is the distance between two elements. Given a norm in a space with subtraction, we can obtain a distance function: the metric induced by a norm.

Figure 1. Hierarchy of topological, metric, normed and inner product spaces. Source.

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

L^{p}

is complete but Riemann isn't.

Normed vector space

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, $∣ x ∣$ )

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.