I'm getting a little confused with the notation for spanning sets. In our notes we have the following definition.
Let $S$ be a vector space over the field $F$. Then:
$\operatorname{span}S = \left\{\sum\limits_{i=1}^n a_iv_i:a_i \in F, v_i \in S\right\}$
We say that $S$ spans $V$ if $V = \operatorname{span}S$
From this definition it seems that a spanning set is the set of all vectors in a vector space. Later in the notes we then say that the set $ \{(1,0), (0,1) \} $ spans $\mathbb{R}^2$. I understand why that set spans $\mathbb{R}^2$ but am getting a bit confused by the different ways span is used. Is $\operatorname{span}S$ the set of all possible vectors in the vector space $S$, hence equivalent to $S$?
Also, I'm asked to determine if $\{1+x, x^2 \}$ spans $P_2(\mathbb{R})$. I'm thinking no because there is no way to get a polynomial such that $x$ and $x^0$ have different coefficients. Is this correct?