What does the phrase "by a polynomial in a finite number of functions of $\mathcal{F}$ means? In Royden's "Real Analysis" (second edition), there is the following exercise concerning the Stone-Weierstrass Theorem (it can be found on page 175, Chap. 9, Sec. 7):

33. Let $\mathcal{F}$ be a family of continuous real-valued functions on a compact Hausdorff space X, and suppose that $\mathcal{F}$ separates the points of X. Then every continuous real-valued function on X can be uniformly approximated by a polynomial in a finite number of functions of $\mathcal{F}$.

I am trying to understand what does the phrase "a polynomial in a finite number of functions of $\mathcal{F}$" menas in this context, but I am failing to do so. I would greatly appreciate if anyone can enlighten me with this one (not with the solution, just with an explanation of what this phrase means).
Also, this problem appears as 44 on Royden's "Real Analysis" (third edition) in Chap. 9, Sec.9 and as 33 on Royden's "Real Analysis" (fourth edition) in Chap. 12 Sec. 12.3.
 A: A real polynomial in $k$ variables is a function of variables $X_1,X_2,\ldots,X_k$ that can be expressed as a linear combination of products of these variables, with real coefficients.  The general definition is therefore a finite sum of the form
$$
  p(X_1,X_2,\ldots,X_k) = \sum a_{n_1,n_2,\ldots,n_k}X_1^{n_1}X_2^{n_2}\ldots X_k^{n_k},
  $$
where each index (and exponent) $n_1,n_2,\ldots,n_k$ can vary from 0 up.
If $f_1,  f_2,  \ldots , f_k$ are members of your family $\mathcal F$,  then you can compose them with a polynomial such as the one above,
obtaining  a function $F$ defined on  $X$ as follows:
$$
  F(x) =   p\big (f_1(x), f_2(x), \ldots  f_k(x)\big ).
  $$
The function $F$ so defined is what one might call a polynomial in a finite number of functions of $\mathcal{F}$.
Of course,  by choosing other polynomials $p$ and other finite subsets of $\mathcal F$ you will get a whole lot of such
functions.

PS: For the cognoscenti,  I am fully aware that I am adopting a functional interpretation of the concept of a polynomial,  which is
perfectly suited to answer the present question!
