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In approximation theory we think of asking questions like "How well can I approximate function $f$ which has certain properties, using functions from some function space - e.g polynomials degree 2".

I gather than whenever people think of function spaces, they are coming from a functional analysis background/viewpoint, and thinking of these as strictly vector spaces (yes they are often defined as such). This makes sense, as imposing some structure on the space you want to talk about is always going to be necessary, and being a vector space is often a pretty reasonable and useful assumption.

However, I'm curious as to what the state of such questions is when we look for approximations to functions in spaces which are not vector spaces e.g: $ \{sin(k x),k\in \mathbb{N}\} $. (Pretty sure this isn't a vector space..)

Clearly asking "What properties does a completely general set of functions have?" is not useful (in approximation theory), but are there common, or interesting properties you can impose on a set without requiring that it be a vector space?

I apologize for what is probably a very silly question, but I am a little lost as to where to look.

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    Approximate in what sense? You can of course impose a metric without imposing a vector space structure: http://en.wikipedia.org/wiki/Metric_space2012-08-10
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    If you're approximating functions valued in a field, and it looks like you're primarily considering $\mathbb{R}$, then any set of functions admits an inclusion into a vector space-at worst by taking each member of the set as a basis element. Can you think of a sensible reason why I'd say $\sin(x)$ is useful for approimation, but not $2\sin(x)$? Normally for a Fourier decomposition we'd take the functions you propose as a basis, and approximate using elements of their span.2012-08-10
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    You make a good point Kevin. I guess I'm thinking of doing something like approximating f as $f=\sum \phi_i(a_i)$, where $\phi_i(a_i)=sin(a_i x)$. Sorry now that I look at that, I can't even really see what I'm trying to do :S2012-08-10
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    Qiaochu, I guess any typical measure of approximation would still work fine ($L_2$?). The sets of functions I'm thinking of can still be subsets of reasonable function spaces.2012-08-10

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