Geodesic
Smooth renormings of the Lebesgue–Bochner function space $L^1(\mu ,X)$
Marián Fabian1 ; Sebastián Lajara2
1Institute of Mathematics Czech Academy of Sciences Žitná 25 115 67 Praha 1, Czech Republic
2Departamento de Matemáticas Escuela de Ingenieros Industriales Universidad de Castilla-La Mancha Campus Universitario 02071 Albacete, Spain
Studia Mathematica, Tome 209 (2012) no. 3, pp. 247-265
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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We show that, if $\mu$ is a probability measure and $X$ is a Banach space, then the space $L^1(\mu,X)$ of Bochner integrable functions admits an equivalent Gâteaux (or uniformly Gâteaux) smooth norm provided that $X$ has such a norm, and that if $X$ admits an equivalent Fréchet (resp. uniformly Fréchet) smooth norm, then $L^1(\mu,X)$ has an equivalent renorming whose restriction to every reflexive subspace is Fréchet (resp. uniformly Fréchet) smooth.
DOI : 10.4064/sm209-3-4
Keywords: probability measure banach space space bochner integrable functions admits equivalent teaux uniformly teaux smooth norm provided has norm admits equivalent chet resp uniformly chet smooth norm has equivalent renorming whose restriction every reflexive subspace chet resp uniformly chet smooth

Marián Fabian  1   ; Sebastián Lajara  2

1 Institute of Mathematics Czech Academy of Sciences Žitná 25 115 67 Praha 1, Czech Republic
2 Departamento de Matemáticas Escuela de Ingenieros Industriales Universidad de Castilla-La Mancha Campus Universitario 02071 Albacete, Spain 
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Marián Fabian; Sebastián Lajara. Smooth renormings of the Lebesgue–Bochner
 function space $L^1(\mu ,X)$. Studia Mathematica, Tome 209 (2012) no. 3, pp. 247-265. doi: 10.4064/sm209-3-4

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