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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.2011-11-26
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    @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.2011-11-26
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    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.2011-11-26
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    @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).2011-11-26
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    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?)2011-11-26
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    I have the feeling that Frank is looking for something like an affine space over a normed vector space, but I can't be sure.2011-11-26
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    @DanielMcLaury Given Strings "ab" and abc", one can do append("ab","c") to transform "ab" into "abc". Obviously replace, prepend, swap, etc. are other options. The space transforms seems to be a vector space for the set of string; the complexity of the transforms (e.g. the number of operations required) can be a norm. Then I can determine the distance between two strings by associating a specific range of transforms with the set of strings. I am asking what the name for the mapping (or the process, I guess) of defining / deriving such corresponding spaces is.2011-11-29

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