Does a given Banach space have any non-trivial complemented subspaces? Usually, the answer is: yes, quite a lot. Sometimes the answer is: no, none at all.
@article{urn:eudml:doc:38635,
title = {Complemented and uncomplemented subspaces of Banach spaces.},
journal = {Extracta Mathematicae},
volume = {15},
year = {2000},
pages = {335-371},
zbl = {0981.46022},
mrnumber = {MR1823896},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:38635}
}
Plichko, Anatolij M.; Yost, David. Complemented and uncomplemented subspaces of Banach spaces.. Extracta Mathematicae, Tome 15 (2000) pp. 335-371. http://gdmltest.u-ga.fr/item/urn:eudml:doc:38635/