Exercise 45, § 1

For every real number $h >0$, let $\Delta_h$ be the continuous function on $\mathbf{R}$ such that $\Delta_h(t) = 0$ if $|t|> h$ and $\Delta_h(t) = 1- |t|/h$ for $t\in [-h, h]$.

a) The family in $\mathscr{C}([0,1])$ formed by the functions $f_0= 1,f_1:t\mapsto t$, and the functions

for $k\geqslant 1$ and $0\leqslant l <2^{k-1}$ is a Banach basis of the Banach space $\mathscr{C}([0,1])$ (cf. EVT, IV, p. 70, exerc. 14). More precisely, for every $f\in$ $\mathscr{C}([0,1])$, one has

$k\geqslant 10\leqslant l<2^{k-1}$

in $\mathscr{C}([0,1])$.

b) The family of functions $t\mapsto e^{2i\pi ht}$ for $h\in \mathbf{Z}$ is not a Banach basis of the space of functions $f\in \mathscr{C}([0,1])$ such that $f(0) =f(1)$.