It is clear that the conclusion of Theorem 1 is no longer necessarily valid if $X$ is no longer assumed to be compact, as is shown by the example of the decreasing sequence of functions $x/(n + x)$ in $\mathbf{R}_+$.

The condition is clearly necessary; let us show that it is sufficient. For each $\varepsilon > 0$, we shall show that there is a function $g \in H$ such that $|f(z) - g(z)| < \varepsilon$ for all $z \in X$. Let $x$ be any point of $X$, and let $H_x$ be the set of all functions $u \in H$ such that $u(x) < f(x) + \varepsilon$. By hypothesis, for each $y \in X$, the function $u_{x, y}$ belongs to $H_x$ and we have $u_{x, y}(y) > f(y) - \varepsilon$. Hence, by Proposition 1, there is a finite number of functions of $H_x$ whose upper envelope $v_x$ is such that $v_x(z) > f(z) - \varepsilon$ for all $z \in X$; on the other hand, we have $v_x(x) < f(x) + \varepsilon$ by the definition of $H_x$; finally, $v_x \in H$ by hypothesis. Proposition 1 therefore shows that there exists a finite number of functions $v_{x_i}$ whose lower envelope $g$ is such that $g(z) < f(z) + \varepsilon$ for all $z \in X$; but since we have $v_{x_i}(z) > f(z) - \varepsilon$ for all $z \in X$ and for every index $i$, we have also $g(z) > f(z) - \varepsilon$ for all $z \in X$. Since $g \in H$ by hypothesis, the proof is complete.

It is enough to show that H satisfies the conditions of the Corollary to Proposition 2. By hypothesis, if $u \in H$ and $v \in H$, we have

On the other hand, let x and y be any two distinct points of X, and let α, β be any two real numbers. By hypothesis, there is a function $h \in H$ such that $h(x) \neq h(y)$: say $h(x) = \gamma$ and $h(y) = \delta$. Since the constants belong to H, the function

An equivalent statement of the theorem is that any subalgebra of $\mathscr{C}(X; \mathbf{R})$ which contains the constant functions and separates the points of X is dense in $\mathscr{C}(X; \mathbf{R})$.

Let $H_0$ be the set of all polynomials in the functions of H, and let $\overline{H}_0$ be the closure of $H_0$ in $\mathscr{C}$. If g is any polynomial in n variables with real coefficients, then $(u_1, u_2, \ldots, u_n) \to g(u_1, u_2, \ldots, u_n)$ is a continuous mapping of $\mathscr{C}^n$ into $\mathscr{C}$, which maps $H_0^n$ into $H_0$, and therefore maps $\overline{H}_0^n$ into $\overline{H}_0$ (Chapter I, § 2, no. 1, Theorem 1). In particular, $\overline{H}_0$ is a vector subspace of $\mathscr{C}$ and evidently satisfies the first and third conditions of Theorem 2; we shall show that it also satisfies the second condition, and this will prove that $\overline{H}_0 = \mathscr{C}$.

Since every function $u \in \overline{H}_0$ is bounded in $X$, it is enough to prove the following lemma:

It is enough to prove the lemma for an interval of the form $I = [-a, +a]$ and hence, replacing $t$ by $at$, for the interval $I = [-1, +1]$. Since $|t| = \sqrt{t^2}$, Lemma 1 is a consequence of the following result:

To prove Lemma 2 it is enough to show that, for all $t \in [0, 1]$, we have

We prove (2) by induction on $n$. It is true for $n = 0$. If $n \geq 0$ it follows from the inductive hypothesis (2) that $0 \leq \sqrt{t} - p_n(t) \leq \sqrt{t}$, hence $0 \leq p_n(t) \leq \sqrt{t}$, and therefore from (1) we have

If $X$ is not compact, the conclusion of Theorem 3 is not necessarily valid. For example, a continuous real-valued function on $\mathbf{R}$ which is bounded and not constant cannot be uniformly approximated in $\mathbf{R}$ by polynomials (cf. Exercise 6).

For if $x = (x_i)$ and $y = (y_i)$ are two distinct points of $X$, there is at least one index $i$ such that $x_i \neq y_i$; hence the family of continuous functions $\mathrm{pr}_i$ satisfies the conditions of Theorem 3.

Consider first the particular case in which $A$ consists of a single point $x_0$. The hypotheses then imply that $H$ separates the points of $X$; for, if $x \neq x_0$, then by hypothesis there is a function $u \in H$ such that $u(x) \neq 0 = u(x_0)$. Hence, for each $\varepsilon > 0$ and each continuous real-valued function $f$ on $X$ such that $f(x_0) = 0$, there exists (Theorem 3) a polynomial $g$ in the functions of $H$ such that $|f(x) - g(x)| \leq \varepsilon$ for all $x \in X$. In particular $|g(x_0)| \leq \varepsilon$, so that

for all $x \in X$; and since $g(x) - g(x_0)$ is a polynomial in the functions of $H$ with no constant term, the result is established in this case.

In the general case, consider the equivalence relation $R$ on $X$ whose classes are the set $A$ and the sets $\{x\}$ for $x \notin A$. The quotient space $X/R$ is Hausdorff (Chapter I, § 8, no. 6, Proposition 15) and therefore compact. Let $\varphi : X \to X/R$ be the canonical mapping. Every continuous real-valued function $f$ on $X$ which vanishes on $A$ can be written in the form $f = f_1 \circ \varphi$, where $f_1$ is a continuous real-valued function on $X/R$ which vanishes at the point $x_0' = \varphi(A)$. Applying the result already proved to the space $X/R$ and the point $x_0'$, we obtain the final result.

Consider the set $H$ of "functions of one variable" $(x_i) \to u_\alpha(x_\alpha)$ (any $\alpha \in I$) which are continuous on $X$. This set separates the points of $X$, for if $x = (x_i)$ and $y = (y_i)$ are any two distinct points of $X$, there exists $\alpha \in I$ such that $x_\alpha \neq y_\alpha$ and there exists a continuous real-valued function $h_\alpha$ on $X_\alpha$ such that $h_\alpha(x_\alpha) \neq h_\alpha(y_\alpha)$. The function $x \to h_\alpha(\operatorname{pr}_\alpha x)$ then belongs to $H$ and takes distinct values at $x$ and $y$. Since every polynomial in the functions of $H$ is of the form stated in the theorem, the result follows from Theorem 3.

If not all the $X_i$ are compact, the conclusion of Theorem 4 is not necessarily valid (cf. Exercise 9).

Given any real number $\varepsilon > 0$, for each $x \in X$ there exists an open neighbourhood of $x$ in which the oscillation of $f$ is $\leq \varepsilon$. Hence there is a finite open covering $(A_i)_{1 \leq i \leq n}$ of $X$ such that the oscillation of $f$ in each $A_i$ is $\leq \varepsilon$. Let $a_i$ be a value of $f$ in $A_i$ ($1 \leq i \leq n$), and let $(u_i)_{1 \leq i \leq n}$ be a continuous partition of unity subordinate to the covering $(A_i)$ (Chapter IX, § 4, no. 4, Corollary to Proposition 4). Let $x$ be any point of $X$. For each index $i$ such that $x \notin A_i$, we have $u_i(x) = 0$, and for each index $i$ such that $x \in A_i$ we have $\|f(x) - a_i\| \leq \varepsilon$; it follows that

On the other hand, by hypothesis there is a function $v_i \in H$ such that

for all $x \in X$ ($1 \leq i \leq n$); hence we have

and the proof is complete.

From Proposition 5 it follows that, to each of the propositions in which we have proved that a certain subset $H$ of $\mathscr{C}(X; \mathbf{R})$ is dense, there corresponds an analogous proposition for continuous mappings of $X$ into an arbitrary normed space $Y$. We shall write down explicitly only the proposition which corresponds in this way to Theorem 3. Given a set $H$ of real-valued functions on $X$, a polynomial in the functions of $H$, with coefficients in $Y$, is defined to be any linear combination, with coefficients in $Y$, of products of a finite (possibly empty) family of functions belonging to $H$. Then:

From this we deduce:

We have only to note that $Y$ is also a normed space over $\mathbf{R}$ and to apply Proposition 6 to the set of real parts and imaginary parts of the functions $f \in H$, using the formulas

We shall see later that in general it is not possible to approximate $f$ uniformly by polynomials (with coefficients in $Y$) in the variables $z_k$ alone, even if $Y = \mathbf{C}$.

If $X'$ is the compact space obtained by adjoining a point at infinity $\omega$ to $X$, then $C_0(X)$ can be identified with the subspace of $C(X; \mathbf{C})$ consisting of continuous mappings which vanish at $\omega$, the norm on $C_0(X)$ being defined by

By virtue of Proposition 7, every $f \in C_0(X)$ can be uniformly approximated by polynomials with complex coefficients in the functions belonging to $A$; moreover, since $f(\omega) = 0$, the argument of no. 2, Proposition 4 shows that we may suppose these polynomials to have no constant term, and then they belong to $A$.

Another application of Proposition 7 is the following:

We have only to observe that P (endowed with the topology of uniform convergence) is canonically isomorphic to the space of all continuous mappings of the compact space $\mathbf{T}^m$ into $\mathbf{C}$ (Chapter VII, § 1, no. 6), and apply Proposition 7 to the set of mappings of $\mathbf{T}^m$ into $\mathbf{C}$ which correspond to the m mappings $(x_1, x_2, \ldots, x_m) \to e(x_i) \quad (1 \leq i \leq m)$ of $\mathbf{R}^m$ into $\mathbf{C}$.