# Euclidean spaces,vector spaces,metric spaces,commodity spaces,Cartesian spaces

What exactly is the difference between Euclidean space,vector space ,metric space ,commodity space,Cartesian space?

I am reading about consumer choice and I came across this line:

"A commodity vector (or commodity bundle ) is a list of amounts of the different commodities and can be viewed as a point in Rl,the commodity space. We can use commodity vectors to represent an individual's consumption levels.The lth entry of the commodity vector stands for the amount of commodity l consumed."

When I googled the meaning of commodity space ,it says - commodity space is Euclidean space and here,as we have taken l entries ,it is l-dimensional Euclidean space.

By l dimensions ,do we mean l directions?

Because when we consider only two commodities in a simple Cartesian plane,we can show them as perpendicular distance from two axis (moving in two perpendicular directions).

Is an l dimensional Cartesian plane similar to l dimensional Euclidean plane?

Since we are taking consumption bundles as commodity vectors,commodity space should be same as vector space.

We are now left with metric space ,the Google search engine defines it as " a set for which distances between all members of the set are defined"

We read about various consumption sets defined as a subset of the commodity space Rl.These consumption sets can be open as well as closed..

This shows that all the spaces listed above are interconnected..or similar to each other.

## 1 Answer

I will try to do this at an intuitive level without being technical. If you want rigorous definitions of any of this I can edit my post.

We need a way to mathematically model the consumption choices that an agent faces. To accomplish this, we use $\mathbb{R}^L_+$ to denote the agent's level of consumption of each of 1,2,..,L goods.

Vectors tie into this because we want to think about how much of each good an agent chooses when optimizing over her preferences. So we don't just think about how much of good $i$ the agent wants. We want to think about how much of good 1 the agent wants, good $2$ he wants, etc. simultaneously. So, there are tons of different "mixes" of these goods that an agent can afford (that belong to this agent's budget set given some prices and wealth) and one of those "mixes" (vectors) represents how much of each of the L goods an agent chooses to consume when maximizing his or her representative utility.

Now - Cartesian space. You should actually discuss the Cartesian coordinate system on Euclidean space. We use Cartesian coordinates to denote a point in Euclidean space. We could, for example, instead use polar coordinates to denote points in Euclidean space.

A metric space is a bit more complicated but you can think of it (this is very simplified) as a space where the distance between any two points is well defined. The distances themselves are called a metric. Having a metric space allows us to use topological methods (for example, the concept of open balls when discussing the local non-satiation of preferences). The most common example of a metric space is three-dimensional Euclidean space.

• Are there any computer graphics available which can help us visualise all these spaces because that can help us understand the concept better.Visualising N commodity vectors having n entries in an N -dimensional Euclidean space is pretty difficult.Computer graphics on this topic must be used by firms.Herein comes the role of graphic designers. – Glitteringstar May 14 '16 at 2:30