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/