Geodesic
The existence of a~best approximating element in $(F)$-space with integral metric
Matematičeskie zametki, Tome 8 (1970) no. 5, pp. 583-594

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The existence is proved of a best approximation element in finite-dimensional subspaces of linear metric spaces of a class containing, in particular, the space $S[0,1]$ of measurable functions. 
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     author = {A. L. Garkavi},
     title = {The existence of a~best approximating element in $(F)$-space with integral metric},
     journal = {Matemati\v{c}eskie zametki},
     pages = {583--594},
     publisher = {mathdoc},
     volume = {8},
     number = {5},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_5_a5/}
}
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A. L. Garkavi. The existence of a~best approximating element in $(F)$-space with integral metric. Matematičeskie zametki, Tome 8 (1970) no. 5, pp. 583-594. http://geodesic.mathdoc.fr/item/MZM_1970_8_5_a5/