What does the notation $\mathbb R[x]$ mean?

I thought it was just the set $\mathbb R^n$ but then I read somewhere that my lecturer wrote $\mathbb R[x] = ${$\alpha_0 + \alpha_1x + \alpha_2x^2 + ... + \alpha_nx^n : \alpha_0, ..., \alpha_n \in \mathbb R$}

Edit: The reason why I asked this question was because I had a tutorial question that said:

Check whether a system {$v_1,...,v_m$} of vectors in $\mathbb R^n$ (in $\mathbb R[x]$) is linearly independent.

I just assumed it meant the same thing when they put it in brackets like that. Since it isn't the case, how must I interpret this question.

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    $\begingroup$ Add $n\in \mathbb N$ to the RHS of $\color{red}\colon$ in the second definition and you have the correct definition. $\endgroup$ – Git Gud Apr 5 '15 at 8:38
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    $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ – Emilio Novati Apr 5 '15 at 8:40
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    $\begingroup$ $\mathbb R[x]$ couldn't possible mean $\mathbb R^n$. The notation $\mathbb R^n$ includes a variable $n$ which appears nowhere in the notation $\mathbb R[x]$. $\endgroup$ – Jack M Apr 5 '15 at 8:40

$\mathbb{R}[x]$ denotes the set of all polynomials with coefficients in $\mathbb{R}$. In particular, this set forms a ring under polynomial addition and multiplication. There is no restriction on the degrees of these polynomials, however, as your post suggests. As GitGud stated in the comments, you need an $n \in \mathbb{N}$ somewhere after the colon in your set builder notation. In particular, note the difference between: $$\{a_0 + a_1x + a_2x^2 + ... + a_nx^n \ | \ a_0, ..., a_n \in \mathbb{R} \}$$ and $$\{a_0 + a_1x + a_2x^2 + ... + a_nx^n \ | \ a_0, ..., a_n \in \mathbb{R} \ \wedge \ n \in \mathbb{N} \}$$

In general, $R[x]$ denotes the set of all polynomials with coefficients in a ring $R$. Common examples include $\mathbb{Q}[x]$ (rational coefficients) and $\mathbb{Z}[x]$ (integer coefficients).

  • $\begingroup$ Is there a common convention for denoting restrictions to the degree of polynomials in this notation? Eg. I want only real polynomials of at most degree 3, then prior I wrote $\mathbb{R}^3[x]$, but I am not sure if this is a widespread notation? What about if I wish to restrict my polynomials to the $[0,1]$ interval? $\endgroup$ – Bence Racskó Apr 5 '15 at 8:45
  • $\begingroup$ None that I am aware of, but I could be wrong. For the former, I'd probably write $\{f(x) \ | \ \deg(f) \leq 3 \ \vee \ f(x) \in \mathbb{R}[x] \}$. The latter you'd probably just use plain English to explain what you want. $\endgroup$ – Kaj Hansen Apr 5 '15 at 8:48
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    $\begingroup$ Some use notations such as $\mathbb R[x]_{\le 3}$ for degree limitations. - Note that restricting to an interval makes no sense as we are talking about polynmoials (expressions with certain coefficients) not polynomial functions (defined on some domain) $\endgroup$ – Hagen von Eitzen Apr 5 '15 at 8:49
  • $\begingroup$ Thanks for the information @HagenvonEitzen. And I guess I just assumed that he was referring to polynomial functions, but that is a good point. +1. $\endgroup$ – Kaj Hansen Apr 5 '15 at 8:54
  • $\begingroup$ Thanks for the responses. That might have been an abuse of notation, but I've seen $\mathbb{R}[x]$ used to denote polynomial functions, and that is what I imagined when I asked my question. $\endgroup$ – Bence Racskó Apr 5 '15 at 9:07

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