A Tour of Mathematical Space
Posted on September 28, 2016.
If you continue to study mathematics past calculus, you will be bombarded by a zoo of mathematical spaces.
The word space by itself is never really defined; it can be taken to be a synonym for set. Typically, a space consists of a set of points along with some kind of structure. That structure can come in many different forms.
Algebraic spaces
There are a lot of different algebraic structures we could give a set. Despite lacking the term space in their names, groups, rings, fields, and related objects certainly satisfy the definition above. For our purposes, we will start with the vector space. Vector spaces (sometimes also called linear spaces) consist of a field K along with a set of vectors V. We often say V is a vector space over K.