Geodesic
A Non-Reflexive Banach Space has Non-Smooth Third Conjugate Space
Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 117-119

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J. Dixmier has observed [3, p. 1070] that a non-reflexive Banach space has non-rotund fourth conjugate space. It is the aim of this paper to improve Dixmier’s result by showing that a non-reflexive Banach space already has non-smooth third conjugate space in that the images under natural embedding of the continuous linear functionals which do not attain their norm on the unit sphere are non-smooth points of the third conjugate space. 
Giles, J. R. A Non-Reflexive Banach Space has Non-Smooth Third Conjugate Space. Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 117-119. doi: 10.4153/CMB-1974-022-0
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