Open ball contains infinitely many disjoint open balls of equal radius

Question

Let $X$ be an infinite dimensional Banach space. Show that the unit ball has contains infinitely many open balls all of equal radius.

The hint is to use Riesz's lemma.

Answer 1

(I may have done mistakes, but it seems to work)

Take $r=1/2$ and $r^\prime=1/4$.

First step : We choose $u_1$ a norm $1$ vector and let $F_1=Span(u_1)$.

Second step : By Riesz's lemma, there exists $u_2$ a $1$ norm vector such that $d(u_2,F_1) \geqslant r$. Then $B(u_1,r^\prime) \cap B(u_2,r^\prime) = \emptyset$. Let $F_2=Span(u_2,F_1)$.

...

Continuing, we obtain an infinite number of vectors $u_1$, ..., of norm $1$ such that $\forall i \neq j, B(u_i,r^\prime) \cap B(u_j,r^\prime) = \emptyset$. Finally take the vectors $v_i=u_i/2$ and consider the balls $B(v_i,r^\prime/2)$ so that everything belongs to the unit ball.