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Kannan maps in normed spaces

Delfina RouxClemente Zanco — 1978

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Si introduce la nozione di «struttura quasi-normale relativa» per i sottoinsiemi chiusi (non necessariamente convessi) di uno spazio normato. Si prova quindi che ogni mappa di Kannan (generalizzata) che muta in sè un sottoinsieme dotato di tale struttura e debolmente compatto di uno spazio normato ha un punto fisso. Analogo risultato vale per i sottoinsiemi dotati di tale struttura e debolmente* chiusi di uno spazio duale; in particolare ogni mappa di Kannan che muta in sè l 1 o L p , 1 < p , (o una sua bolla...

Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces

Victor KleeLibor VeselýClemente Zanco — 1996

Studia Mathematica

For combining two convex bodies C and D to produce a third body, two of the most important ways are the operation ∓ of forming the closure of the vector sum C+D and the operation γ̅ of forming the closure of the convex hull of C ⋃ D. When the containing normed linear space X is reflexive, it follows from weak compactness that the vector sum and the convex hull are already closed, and from this it follows that the class of all rotund bodies in X is stable with respect to the operation ∓ and the class...

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