Geodesic
On complex strictly convex complexifications of Banach spaces
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 7 (2000) no. 3, pp. 299-307 Cet article a éte moissonné depuis la source Math-Net.Ru

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We show that every real separable normed space may be complexified to a complex strictly convex normed space. The same result is obtained also for some classes of nonseparable spases, for example, for spases $X$ with 1-norming separable subspases in $X^*$; however, a space $\ell_\infty(\Gamma)$ has no complex strictly convex complexifications. 
@article{JMAG_2000_7_3_a3,
     author = {V. M. Kadets and A. Yu. Kellerman},
     title = {On complex strictly convex complexifications of {Banach} spaces},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {299--307},
     year = {2000},
     volume = {7},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2000_7_3_a3/}
}
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V. M. Kadets; A. Yu. Kellerman. On complex strictly convex complexifications of Banach spaces. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 7 (2000) no. 3, pp. 299-307. http://geodesic.mathdoc.fr/item/JMAG_2000_7_3_a3/