Exercise 22, § 4

Let X be a real Banach space, E a finite dimensional subspace of X. a) Let S be the unit sphere in X. Show that for every $\varepsilon > 0$, there exists a finite number of linearly independent points $z_i \in S$ ($1 \leq i \leq r$) such that for every $x \in S \cap E$, there exists an index $i$ such that $\|x - z_i\| \leq \varepsilon$. b) Let $z'_i$ ($1 \leq i \leq r$) be points in the dual $E'$ of E with $\|z'_i\| \geq 1$ for every $i$, and $\langle z_i, z'_j \rangle = \delta_{ij}$ (Kronecker’s index), and let F be the closed subspace of codimension $r$ in E which is orthogonal to the subspace of E' generated by the z'_i. Show that for every x ∈ S ∩ E and for every y ∈ F, \|x + y\| ≥ 1 − ε, and in particular, E ∩ F = {0}, so that the sum E + F is the topological direct sum. c) Deduce from b) that the continuous projector P from E + F onto E corresponding to the topological direct sum decomposition E ⊕ F has a norm \|P\| < 1/(1 − ε).