4
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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?

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  • $\begingroup$ You may find this interesting (same question), especially the comment section. bests $\endgroup$ – user190080 Jun 22 '15 at 13:03
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    $\begingroup$ 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. $\endgroup$ – goblin Jun 22 '15 at 14:37
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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.

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