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A follow up to this question. I kinda understand that the vector space of polynomials (P(R)) is a subspace to the infinite set of numbers between the interval [0,1]. Is this because we can represent an arbitrary number within the interval with polynomial of any degree? Something I am missing here.

Justin
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    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Oct 22 '22 at 17:25
  • I'm not sure how to further clarify the question. I'm guess i am trying to better understand the relationship between polynomials and the infinite interval [0,1]. @Community – Justin Oct 22 '22 at 17:34
  • Start by trying to understand what this vector space of "all continuous, real-valued functions on $[0,1]" is. What are the elements? What are the operations? Addition? Scalar multiplication? –  Oct 22 '22 at 18:13
  • The set of polynomials with real coefficients on $[0,1]$ is a subspace of the real-valued continuous functions on $[0,1]$, $C([0,1])$. – M A Pelto Oct 22 '22 at 18:25
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    The fundamental theorem of algebra implies that the sequence $p_n(t)=t^n$, $n\in\mathbb{Z}_+$ is linearly independent. – Mittens Oct 22 '22 at 21:35

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