Geodesic
A new relationship between decomposability and convexity
Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 457-466 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In the present work we prove that, in the space of Pettis integrable functions, any subset that is decomposable and closed with respect to the topology induced by the so-called Alexiewicz norm $\left| \left\|\cdot \right\| \right|$ \big(where $\left| \left\| f\right\| \right| =\sup_{[a,b] \subset [0,1]} \big\| \int_{a}^{b}f(s) ds \big\|$\big) is convex. As a consequence, any such family of Pettis integrable functions is also weakly closed.
In the present work we prove that, in the space of Pettis integrable functions, any subset that is decomposable and closed with respect to the topology induced by the so-called Alexiewicz norm $\left| \left\|\cdot \right\| \right|$ \big(where $\left| \left\| f\right\| \right| =\sup_{[a,b] \subset [0,1]} \big\| \int_{a}^{b}f(s) ds \big\|$\big) is convex. As a consequence, any such family of Pettis integrable functions is also weakly closed.
Classification : 28B05, 46A20, 46A55, 46E30, 46E40, 46G10, 52A07, 54A10
Keywords: Pettis integral; decomposable set; convex set; Alexiewicz norm 
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     author = {Satco, Bianca},
     title = {A new relationship between decomposability and convexity},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2006_47_3_a7/}
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Satco, Bianca. A new relationship between decomposability and convexity. Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 457-466. http://geodesic.mathdoc.fr/item/CMUC_2006_47_3_a7/