Davie's construction of subspaces of $c_0$ and $\ell_p$ ($p\in (2, \infty)$) without the approximation property, as outlined in Section 2.d of Lindenstrauss and Tzafriri's book Classical Banach Spaces I, uses a probabalistic lemma (Lemma 2.d.4, p.87-88). I do not know Davie's proof all that intimately, having been through it only once - courtesy of a fellow grad student who took a couple of hours to go over it in a research group seminar... I remember that it looked like magic at the time.

Davie's construction of subspaces of $c_0$ and $\ell_p$ ($p\in (2, \infty)$) without the approximation property, as outlined in Section 2.d of Lindenstrauss and Tzafriri's book Classical Banach Spaces I, uses a probabilistic lemma (Lemma 2.d.4, p.87-88). I do not know Davie's proof all that intimately, having been through it only once - courtesy of a fellow grad student who took a couple of hours to go over it in a research group seminar... I remember that it looked like magic at the time.

(Edited once for a typo)

Davie's construction of subspaces of $c_0$ and $\ell_p$ ($2<p<\infty$), as outlined in Section 2.d of Lindenstrauss and Tzafriri's book Classical Banach Spaces I uses a probabalistic lemma (Lemma 2.d.4, p.87-88). I do not know Davie's proof intimately, having only been through it only once courtesy of a fellow grad student who took a couple of hours to go over it in a research group seminar. I remember that it looked like magic at the time (the proof as a whole, not just the above-mentioned lemma).

Davie's construction of subspaces of $c_0$ and $\ell_p$ ($p\in (2, \infty)$) without the approximation property, as outlined in Section 2.d of Lindenstrauss and Tzafriri's book Classical Banach Spaces I, uses a probabalistic lemma (Lemma 2.d.4, p.87-88). I do not know Davie's proof all that intimately, having been through it only once - courtesy of a fellow grad student who took a couple of hours to go over it in a research group seminar... I remember that it looked like magic at the time.