Exercise 19, § 4

Let E be the subspace of the Banach space $C([ - 1, 1]) ; \mathbf{R}$ of all continuous mappings from $[-1, 1]$ into $\mathbf{R}$, consisting of the polynomials. Similarly, let F be the subspace of $C([0, 1]) ; \mathbf{R}$ consisting of all polynomials. Let $u$ be the mapping which associates to each polynomial $f \in E$, the polynomial $t \mapsto \frac{1}{2}(f(\sqrt{t}) + f(-\sqrt{t}))$ in F. Also, let $w$ be the mapping from E into F which associates with every polynomial $f \in E$ its restriction to $[0, 1]$. Show that $u$ is a strict morphism from E onto F, but that for every $\varepsilon > 0$, $u + \varepsilon w$ is not a strict morphism from E into F.