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The question is pretty much in the title. I have been using $\mathrm{poly}_p$ but I don't like it.

fred
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2 Answers2

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The usual notation that I have seen extensively used is $\mathbb{R}_p[X]$.

Clement C.
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    Thanks, what does the $X$ mean? Is that the domain? – fred Jan 31 '16 at 19:21
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    It's the formal variable of the polynomial (i.e., any $P\in\mathbb{R}p[X]$ is of the form $\sum{n=0}^p a_n X^n$, with $a_n\in\mathbb{R}$ for all $0\leq n \leq p$). – Clement C. Jan 31 '16 at 19:24
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    Further note: although I have seen this notation used in English as well during my studies (and in some articles), the only "Wikipedia reference" I can find is in French. See also (again in French) these notes. – Clement C. Jan 31 '16 at 19:27
  • Hi, am am going to choose the other guy's answer but I really appreciate your help. I am only choosing his because I am an ignorant American and can't read French. I really do appreciate it. – fred Jan 31 '16 at 19:32
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    No worries at all :) – Clement C. Jan 31 '16 at 19:32
  • @Clement C. ,Is there separate notation for the vector space with this set? – Michael Levy Feb 12 '21 at 14:47
  • As I understand it, this denotes the set of "formal" polynomials. How would one denote the set of polynomial functions? I realise these are essentially the same thing (isomorphic), but I want to be explicit that I'm referring to a set of functions. – Anakhand Mar 05 '21 at 12:42
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Wikipedia calls this $P_p$, where the capital $P$ stands for polynomial and the lower case stands for the maximum degree. I have seen this in some textbooks as well, but it's not universal.

This object is somewhat unusual; it is a vector space, but not a ring. It seems strange to have polynomials and not multiply them. It's basically $\mathbb{R}^{p+1}$ but written with plusses and dummy variables, rather than with commas.

vadim123
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  • It's not that unusual from a linear algebra/functional analysis/operator theory point of view: It's a good example to show the relations between a linear operator (e.g. $\frac{d}{dx}$ and a matrix representing the operator. I think it's in particular instuctive because the vector space is not $\mathbb R^n$, although it's isomorphic to it. Naturally, you'd study differential operators in other vector spaces (Sobolev or Schwartz, for example), but it's a good place to start studying differential operators. – Roland Jan 31 '16 at 19:36
  • @vadim123 , is there a separate notation for the vector field with this this set? – Michael Levy Feb 12 '21 at 14:46