# What is the proper term for the entity that relates a vector space and a set?

One way to generate a metric for a set $S$ (a distance function between elements $a,b$ of the set $S$) would be by associating it with a vector space $V$ (the vectors that connect the elements $a,b$) and using the norm (length function) of the vectors as the distance metric for $S$.

What would be the proper term for the entity that establishes the association between the set and the vector space, i.e. for a function $f: (a,b) \Rightarrow V$?

I guess I am looking for the appropriate synonym for a term like "vectorizer" or "vector space associator".

Example: given a set $S$ of strings over some alphabet, one can define a vector space $T$ of string transforms whose elements are mappings $f: (S) \Rightarrow S$. Given a norm (e.g. number of edit operations inside the transform) on $T$ via $n: (T) \Rightarrow R$ we can then induce a metric on $S$. But to do that, we first need a mapping $f: (S, S) \Rightarrow T$ to get from the set $S$ to the vectors $T$. What is the proper term for such a mapping?

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What is the vector space you associate to the set $S$? A natural candidate would be the Hilbert space of all square-summable sequences over the set $S$. Then to $a,b\in S$ you associate the vector that is the difference between the basis elements corresponding to $a$ and $b$. But this is by no means the only choice. You will have to clarify your question. –  Stefan Geschke Nov 26 '11 at 7:12
@StefanGeschke Hm, any suitable vector space, I guess, depending on the set S. Two relevant examples would be Euclidean vectors and transform functions f: a => b. –  Frank Abagnale Nov 26 '11 at 7:15
If you say euclidean vectors, do you consider $S$ to be a subset of some $\mathbb R^n$? I have no idea what you mean by the transform functions. –  Stefan Geschke Nov 26 '11 at 7:17
@StefanGeschke Sorry. On Euclidean vectors: yes. On transform functions: I'm thinking about computations such as string transforms, f : (String,String) => StringTransform, with some associated norm n : (StringTransform) => R (don't know how to do the fancy real-numbers R). –  Frank Abagnale Nov 26 '11 at 7:32
Can you give an example of a set of strings and a vector space of transformations of them? I don't think I understand the question. (For instance, what would it mean to multiply a string transformation by negative pi?) –  Daniel McLaury Nov 26 '11 at 8:58

"Vector space" is a term denoting a certain kind of mathematical structure: A set $S$ provided with an addition and a multiplication by "scalars" such that certain rules hold. The elements of such a space, whatever they are (numbers, strings, functions, etc.) are called vectors. Vectors have nothing to do with arrows per se – it is just a historical coincidence that the first vectors encountered by mathematicians were the forces arrows drawn by physicists.
In your case it seems that in the first place you have arrows representing a map $f: S\to S$. You can use these arrows to create a directed graph $\Gamma$ with vertex set $S$ and the given arrows as edges. When it comes to defining a metric on $S$ one should forget about the direction of these edges to make the distance function symmetric. On the other hand you may assign positive weights to the edges accounting for some (computation) cost.
The graph $\Gamma$ is connected if for any two $x$, $y\in S$ there is an edge-path connecting $x$ and $y$ in the obvious way. Any path connecting $x$ and $y$ has a total cost, and a natural distance function in your setting would be $d(x,y):=$ minimal cost to get from $x$ to $y$.