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

Question

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.

Answer 1

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.

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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.

Answer 2

Of all the types of structures or spaces you mention, vector space $\mathcal{V}$ is the simplest type of structure. Other structures are constructed from this:

1. A real vector space with an inner product (which allows to define the ortogonality relation) is a Hilbert space $\mathcal{H}$.
2. An Euclidean space is an affine space $\mathcal{A}$, a set of points, such that for every pair of points $P$ and $Q$ there is a a vector $\overrightarrow{PQ}$ of a finite dimensional Hilber space. In an Euclidean space you can define, distances, angles, points, etc. so you have a geometrical space.
3. A set of points with a distance is a metric space, Euclidean space is a metric space, but not every metric space is an Euclidean space.
4. A commodity space is a set of points, where you can define a total order relation, which is continuous and has additional properties (some convex sets of the Euclidean spaces can become a commodity space, if you define some additional structure on them).
5. A Cartesian space, is just a set that can be expressed as Cartesian product of sets. Again, mutidimensional Euclidean space and many commodity spaces are particular instances of Cartesian spaces.

Answer 3

You have to distinguish between mathematical concepts and economic concepts.

Vector space, euclidean space, metric space, are mathematical concepts. Commodity space is an economic concept.

The most basic spaces are vector spaces and metric spaces.

A vector space on $R$ is defined as a set $V$ (of elements of any nature) such that there are two operations called scalar multiplication and addition, that satisfy a list of certain properties (too long to be described here).

Vector space is the fundamental concept of linear algebra. The most well-known example of vector space is $R^n$.

A metric space is defined as a set (of elements of any nature) on which is defined a function that is called distance or metric, that satisfies specific axioms (too long to be described here, but you can easily find them on internet).

It is a generalization of the everyday concept of distance.

Vector spaces and metric spaces are two independent concepts: a vector space, on its own, is not a metric space and a metric space, on its own, is not a vector space.

In order to make a vector space a metric space, you have to define on it a distance.

This is the case of an euclidean space, that is a vector space ($R^n$) on which is defined an inner (or scalar) product. It is possible to prove that an inner product induces a distance, so that an Euclidean space is a metric vector space.

An example of metric space that is not a vector space is the so-called discrete space, which is a set (of elements of any nature) on which is introduced a distance called the discrete distance.

The commodity space, instead, is an economic concept, that from a mathematical point of view can be seen as a vector space, in particular $R^n$ if the number of commodities is $n$.

You wrote:

"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”.

That is, the vectors in $R^n$ are interpreted as bundles of goods, but this is an economic interpretation, and not a mathematical concept.

Other properties can be introduced in the concept of commodity bundle, as a relation of total ordering, and others, but these are different from vector space properties.

Now we can answer your previous question:

“By l dimensions ,( of a commodity space) do we mean l directions? “

We saw that, from a mathematical point of view, the commodity space is a vector space, in particular is $R^n$, so we have to relate to the definition of dimension of a vector space. This definition is a little complicated because implies the concept of base of a vector space (too long to be explained here). In a general vector space, the dimension of the space is defined as the number of vectors that constitutes a base.

In $R^n$ (n-ples of real numbers) one can prove that the dimension is the number of real numbers that forms a vector, that is $n$. So, in a commodity space the dimension is the number of goods that constitute a commodity bundle.

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