In the book Symmetry, Representations and Invariants they give the following definition:
If $V$ is a finite dimensional $\mathbb{C}$-vector space, a function $f\colon V\to \mathbb{C}$ is a polynomial of degree $\leq k$ if for some basis $\{e_{1},\dots,e_{n}\}$ of $V$ one has $$f\left (\sum_{i=1}^{n}x_{i}e_{i}\right ) = \sum_{|I|\leq k}a_{I}x^{I}.$$ If there is a multi-index $I$ with $|I|=k$ and $a_{I}\neq 0$, then we say $f$ has degree $k$. If $a_{I}=0$ when $|I|\neq k$, we say $f$ is homogeneous of degree $k$. Let $\mathcal{P}(V)$ be the set of all polynomials on $V$.
Then they say that $\mathcal{P}(V)$ is an algebra, and it is generated by the linear coordinate functions $x_{1},\dots,x_{n}$. I understand that when we fix a basis on $V$, then we have an algebra isomorphism $\mathcal{P}(V)\cong \mathbb{C}[x_{1},\dots,x_{n}]$. My question is a bit silly: what do they mean by coordinate functions and how do they generate this algebra? I was thinking maybe that, given a fix ordered basis $E=(e_{1},\dots,e_{n})$, they mean $x_{i}\colon V\to \mathbb{C}$ and map $x_{i}(x) = x_{i}$, if $x$ is expressed in the $E$ basis, i.e. $[x]_{E}=(x_{1},\dots, x_{n})$. Now, if this functions generates $\mathcal{P}(V)$ then for every polynomial $f$ we can write $f = \sum\alpha_{i}x_{i}$ right? Or am I missing something? This last part about how they generate it is not so clear to me. I would appreciate your help.
Yes, according to your algebra-isomorphism,
$p(x_1,\ldots,x_n)\in\mathbb{C}[x_{1},\ldots,x_{n}]$
sends $\;t_1 e_1+\ldots+t_n e_n\;$ to $\;p(t_1,\ldots,t_n)$.
$\\$
A complex algebra $A$ is said to be generated by $a_1,\ldots,a_n$
iff every element of $A$ can be written as a complex polynomial of $a_1,\ldots,a_n$.
That's the definition of generators of an algebra.
$\mathbb{C}[x_1,\ldots,x_n]$ is the most obvious example of this. Every element can by definition be written as a complex polynomial of $x_1,\ldots,x_n$.