Geodesic
The existence of a~non-hereditarily complete family in an arbitrary separable Banach space
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 274-277

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It is proved that every separable Banach space $E$ contains a complete minimal family $\{x_j\}_1^\infty$ with the total biorthogonal family $\{f_j\}_1^\infty$ (in $E^*$) but not hereditarily complete (this means that the closed linear envelope of the f amily $\{f_j(z)x_j\}_1^\infty)$ does not coincide with $E$). 
@article{ZNSL_1979_92_a19,
     author = {V. I. Gurarii},
     title = {The existence of a~non-hereditarily complete family in an arbitrary separable {Banach} space},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {274--277},
     publisher = {mathdoc},
     volume = {92},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_92_a19/}
}
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V. I. Gurarii. The existence of a~non-hereditarily complete family in an arbitrary separable Banach space. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 274-277. http://geodesic.mathdoc.fr/item/ZNSL_1979_92_a19/