Exercise 13, § 1

Let E be a Banach space of type countable over K. We recall (EVT, IV, p. 70, exercise 14) that a sequence $(x_n)_{n\in\mathbf{N}}$ of elements of E is called a Banach basis of E if, for every $x\in E$, there exists a unique sequence of scalars $(\lambda_n(x))_{n\in\mathbf{N}}$ such that

where the series converges in E.

a) Every Hilbert space of type countable admits a Banach basis; every space $\ell^p$(I), where I is countable, admits a Banach basis.

b) The Banach space E of bounded holomorphic functions on the open disk D of radius 1 in the complex plane, endowed with the norm $\|f\|=$ sup$_{z\in D}|f(z)|$, possesses a Banach basis.

c) Every Banach space E admitting a Banach basis is an approximation space. More precisely, there exists a real number $c\geqslant 0$ such that $1_E$ belongs to the closure in the space $\mathscr{L}_{pc}(E)$ of the set of endomorphisms of finite rank and of norm $\leqslant c$ in E.