Example 4

In the normed space $\mathscr{C}(I; \mathbf{C})$ of functions that are continuous over $I = \{0, 1\}$, the restriction to I of the functions $x^n$ ($n \in \mathbf{N}$) form an algebraically independent family. But there exists a sequence of polynomials $(p_n)$ such that $p_n(x^2)$ converges uniformly to $x$ in I (GT, X, § 4.2, lemma 2) which shows that $x$ belongs to the closed vector subspace of $\mathscr{C}(I; \mathbf{C})$ generated by the functions $x^{2n}$ ($n \in \mathbf{N}$).