Example 2

In the normed space $\mathscr{B}(\mathbf{N})$ of bounded sequences of real numbers (I, p. 4) the extremal points of the unit ball are the points $x = (\xi_n)$ such that $|\xi_n| = 1$ for all $n$. For, suppose that we had $|\xi_n| \leq 1$ for all $n$ and $|\xi_p| < 1$ for one index $p$. We can then write

where $y$ (resp. $z$) is the point all of whose coordinates are equal to the coordinate of $x$ with the same index, except in the case of index $p$ where the coordinate is equal to 1 (resp. $-1$). This shows that $x$ is not extremal, since we have $\|y\| \leq 1$ and $\|z\| \leq 1$. Conversely, if $|\xi_n| = 1$ for all $n$, then $x$ is extremal, for the relation $\xi_n = \lambda \eta_n + (1 - \lambda) \zeta_n$ with $|\eta_n| \leq 1,\ |\zeta_n| \leq 1$ and $0 < \lambda < 1$ implies $\xi_n = \eta_n = \zeta_n$.