symmetric functions vs symmetric polynomials I am doing my thesis related with symmetric functions and representations. 
It is for this reason that I am reading MacDonald's book Symmetric Functions and Hall Polynomials.
When reading chapter 1.2. The ring of symmetric functions, he also talks about symmetric polynomials. And he compares both notions and sometimes treats them as equivalent but sometimes not. Does anyone have a clearer explanation about what is the difference (if there are differences)?
So, what is the difference between symmetric functions and symmetric polynomials?
 A: The ring of symmetric functions is a direct limit of the rings of symmetric polynomials in $n$ indeterminates, $n$ going to infinity (for certain injective morphisms that are not inclusion maps). There is no such thing as a (non constant) symmetric polynomial in infinitely many indeterminates, unless one bends the definitions, because a polynomial can have only finitely many terms. For this reason it is sensible to make the terminological distinction, even though symmetric functions are no more "functions" than polynomials are.
By the way, the ring of symmetric functions can also be defined as an inverse limit of the rings of symmetric polynomials, which is maybe more natural and what Macdonald does, but in that case it is important to take the limit in the category of graded rings, for otherwise the limit would give you more than is desired.

So since the is some questioning about this in the comments, let me detail. In the direct limit construction, the injective morphisms to rings with more indeterminates send each elementary symmetric polynomial to the corresponding elementary symmetric polynomial, which, them being morphisms, determines them. (They also send each orbit sum of monomials of degree no more than the number of variables to the corresponding larger orbit sum; this helps to show that for instance using complete homogeneous symmetric polynomials rather than elementary ones has the same effect. For higher degrees, the images of orbit sums do gather some additional monomials, for instance going from one indeterminate $X$ to $X,Y$, the image of the monomial $X^n=e_1[X]^n$ is not $X^n+Y^n$, but the full binomial expansion $e_1[X,Y]^n=\sum_{k=0}^n\binom nkX^kY^{n-k}$.) The image of this morphism is complementary to the kernel of a surjective morphism in the opposite direction that is restricted from the morphism of full polynomial rings sending new indeterminates to zero and keeping the old ones intact. Such morphisms are the ones used in the inverse limit construction, but as said the inverse limit must be taken in the category of graded rings lest such beasts as $\prod_{i\in\Bbb N}(1+X_i)$ spring into existence, destroying the graded nature of the ring. It is an easy exercise to see that the direct and indirect limit constructions define isomorphic (graded) rings.
Either definition does not actually say what a symmetric function is, just what the ring of all of them looks like, up to isomorphism. If one wants to define the ring of symmetric functions as a substructure of something familiar, that is also possible: one could define them as symmetric formal power series in a countable set of indeterminates with bounded degree terms. This is the point of view taken by for instance Stanley (Enumerative Combinatorics, 2) and Sagan (The symmetric group; Representation, combinatorial algorithms and symmetric functions).
While I'm at it, let me add why I started with the direct limit construction, even though both kinds of limits can be used. The direct limit construction morally corresponds better with the way I think of symmetric functions: every individual computation in the ring can be faithfully represented inside some ring of symmetric polynomials, and as such the ring behaves like a infinite union of subrings, not as an inverse limit construction which usually brings into existence infinite values of which only shadows exist in the finite domain. The inverse limit of graded rings precisely avoids such values, because in every fixed degree the surjective morphisms ultimately become bijections.
