Geodesic
On orthogonalization of bases
Matematičeskie zametki, Tome 6 (1969) no. 6, pp. 713-724

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An example of a basis for space $C$, close to the Schauder system, is constructed which, after orthogonalization by the Schmidt method, is not a basis for space $L^p$ for any $p\in[1,2)+(2,+\infty)$. 
@article{MZM_1969_6_6_a7,
     author = {V. M. Veselov},
     title = {On orthogonalization of bases},
     journal = {Matemati\v{c}eskie zametki},
     pages = {713--724},
     publisher = {mathdoc},
     volume = {6},
     number = {6},
     year = {1969},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1969_6_6_a7/}
}
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V. M. Veselov. On orthogonalization of bases. Matematičeskie zametki, Tome 6 (1969) no. 6, pp. 713-724. http://geodesic.mathdoc.fr/item/MZM_1969_6_6_a7/