Geodesic
An approximative property of spaces of continuous functions
Glasgow mathematical journal, Tome 15 (1974) no. 1, pp. 48-53

Voir la notice de l'article provenant de la source Cambridge University Press

A Banach space X is said to have property (PROXBID) if the canonical image of X in its bidual X** is proximal. In other words, if J: X → X** is the canonical embedding, then it is required that every element of X** have at least one best approximation (i.e., nearest point) from the closed subspace J(X). We show below that, if X is the space of (real or complex) continuous functions on a compact set, or the space of (real or complex) continuous functions that vanish at infinity on a locally compact set, then X has property (PROXBID). At this point we should mention the existence of a variety of examples [2, 8] of Banach spaces which lack property (PROXBID). 
Holmes, R. B.; Ward, J. D. An approximative property of spaces of continuous functions. Glasgow mathematical journal, Tome 15 (1974) no. 1, pp. 48-53. doi: 10.1017/S001708950000210X
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