Polynomial ring as a ring of functions My book describes the polynomial ring $R[X]$ as: $R[X] = R[\mathbb{N}] = \{f: \mathbb{N} \rightarrow R | \hspace{ 2mm} f(n) = 0, n \gg 0\}$. What is exactly meant by this?
What do the double arrows mean $(\gg)$? The author is a Danish mathematician and the book is designed for a first semester algebra course. 
 A: The $>>0$ just means for all sufficiently large natural numbers.
Now how to swallow this confusing definition. Intuitively a polynomial in $R[x]$ is just $a_0 + a_1x + ... + a_n^n$ for some $a_0,a_1,...,a_n \in R$. Of course, this is not a very rigorous definition. So let's analyze this definition. First of all, $f(n)$ is supposed to represent the coefficient on $x^n$ (or maybe $x^{n-1}$ depending on what the convention for $\mathbb{N}$ is in Denmark). So our polynomial is just $P(x) = \sum_{n \ge 0} f(n)x^n$.
The $f(n) = 0, n >> 0$ just means for all sufficiently large $n$ we have $f(n) = 0$. This is to guarantee the polynomial has a bounded degree. So this is basically how this definition works. Its essentially defining the polynomial as a sequence of its coefficients, where for all sufficiently large indices we have the element equals $0$.
A: $\gg$ usually means "far greater", which in most contexts is not a precise statement. In this, it means $\exists N\in\mathbb{N} \forall n\in\mathbb{N}\colon f(n)=0$ or "$f$ eventually vanishes" or, equivalently, "$f$ is non-zero for only finitely many $n\in\mathbb{N}$"
A: Reverse engineering this cryptic piece of notation, I would guess that since what should be intended is that $f$ is $0$ at all but finitely many values, the $\gt\gt$ is actually meant to be $\gg$ (produced by the $\TeX$ command \gg), a symbol that usually means "far greater than" and is being used in an unusual way here, not least since it usually means that the ratio of the quantities is large, whereas there is no ratio in comparison to $0$, so if the symbol was going to be used at all, it should have been $n\gg1$.
A: The author appears to be constructing the polynomial ring as a special case of a monoid ring. See the linked Wikipedia page for details.
