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:
- 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.
- 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?
$\endgroup$ 91 Answer
$\begingroup$Quoting the Wikipedia entry for Function space:
$\endgroup$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.
