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I'm a physics student and I don't have a very strong mathematical background.

When I took the course of group theory in physics, I had trouble in telling the difference between the space spanned by the group elements and the function space that is used to represent the group elements.

The main problem that I have is that what defines a function space? Can it be explained without using too much terminologies?

I have read the answers in this post, but it comes to me as:

  1. A function space can be regarded as a set of functions, and if we are lucky than we can find a set of proper basis functions in this space.
  2. If we know all the basis functions of a function space than we can span the function space.

Have I misunderstood something? And above all, can I regard a function space as a set of functions?

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  • $\begingroup$ A "function space" is simply a set of functions from a given set to another given set. There isn't anything particularly important about the term "space," although most function spaces of interest are e.g. topological spaces or vector spaces. $\endgroup$ – Math1000 Oct 26 '17 at 10:31
  • $\begingroup$ @ Math1000 it is just the part "from a set to another set" that confused me when I learnt it from Wikipedia. For me, a set of functions just looks like {1,$x$, $x^2$}, so how to see that it is something from one set to another set? $\endgroup$ – meTchaikovsky Oct 26 '17 at 10:36
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    $\begingroup$ $1$, $x$, and $x^2$ aren't "functions," without further context. For example, we might have $f_n:\mathbb R\to\mathbb R$ with $f_n(x) = x^n$, $n=0,1,2$. Then the set $\{f_0, f_1, f_2\}$ would be a function space; a subset of the space of all functions from $\mathbb R\to\mathbb R$. $\endgroup$ – Math1000 Oct 26 '17 at 11:00
  • $\begingroup$ @ Math1000 so based on your example can I take it that functions are "lines" that connect two fields? And there are sets that consisting of such "lines" which we call function spaces? If this is the case, then are we guaranteed that we can always find a basis in a function space? $\endgroup$ – meTchaikovsky Oct 26 '17 at 11:06
  • $\begingroup$ "Line," "field," and "basis" have precise (and different) definitions in different branches of mathematics, and a function space need not have to do anything with any of them. Consider $X=\{a,b\}$, $Y=\{\alpha,\beta\}$ and $f,g:X\to Y$ with $f(a)=\alpha$, $f(b)=\beta$, $g(a)=\beta$, and $g(b)=\alpha$. Then $\{f,g\}$ is a (perhaps abstract) function space. $\endgroup$ – Math1000 Oct 26 '17 at 11:28
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Quoting the Wikipedia entry for Function space:

In mathematics, a function space is a set of functions of a given kind from a set $X$ to a set $Y$. It is called a space because in many applications it is a topological space (including metric spaces), a vector space, or both. Namely, if $Y$ is a field, functions have inherent vector structure with two operations of pointwise addition and multiplication to a scalar. Topological and metrical structures of function spaces are more diverse.

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