# $\operatorname{span}(x^0, x^1, x^2,\cdots)$ and the vector space of all real valued continuous functions on $\Bbb R$

Let $p_n(x)=x^n$ for $x\in\Bbb R$ and let $\mathcal P=span\{p_0,p_1,p_2,\cdots\}$ . Then

1. $\mathcal P$ is the vector space of all real valued continuous functions on $\Bbb R$.
2. $\mathcal P$ is the subspace of all real valued continuous functions on $\Bbb R$.
3. $\{p_0,p_1,p_2,\cdots\}$ is a linearly independent set in the vector space of all continuous functions on $\Bbb R$.
4. Trigonometric functions belong to $\mathcal P$

Not 1 since sinc function is real valued and continuous, and not 4, too. Can I conclude 2,3 are the correct ones?

• You may find this interesting (same question), especially the comment section. bests – user190080 Jun 22 '15 at 13:03
• 2. is not correct because there isn't one subspace of all real valued functions, there a many. So $\mathcal{P}$ is a subspace, but its not the subspace. – goblin Jun 22 '15 at 14:37

Exactly.
4. doesn't hold in general, so neither does 1.
2. and 3. are true, $\mathcal P$ is a subspace of the space of all continuous functions, and the polynomials $p_n$ are linearly independent from each other.