Basis of all real polynomials?

Question

I am studying the book Topics in Algebraic Graph Theory by Beineke et all and the page 12. By the book, the set of all real polynomials can be generated by the set $\{1,x,x^2,\ldots\}$ which I understand such that

$$\mathbb{R}[x]=\langle1,x,x^{2},\ldots\rangle$$

where I want to make sure that I have understood the notation correctly.

Is my notation $\mathbb R[x]$ correct for all real polynomials? So $\{1,x,x^2,\ldots\}$ spans $\mathbb R[x]$?

Examples. Correct?

1. Which vector space does the $\{ x \}$ span? Not vector space, multiplicative identity not in $\{ x\}$.

2. Size changing like this?

   2. $\langle 1 \rangle=\mathbb R$ is the smallest vector space while

   3. $\langle 1,x\rangle$ is a little bit larger, then

   4. $\langle 1,x,x^2,\ldots\rangle=\mathbb R[x]$ would be the largest.

3. $x^2\not\in\langle 1,x,x^{11}\rangle$

4. $\langle 1,x\rangle=\mathbb R[x] / \langle 1, x^2,x^3,\ldots\rangle$ (the first vector space complemented with the second)?