On the Geometry of Müntz Spaces. (16th June 2015) - Ludkovsky, Sergey V.; Lusky, Wolfgang
Abstract : LetΛ = { λ k } k = 1 ∞ satisfy0 < λ 1 < λ 2 < ⋯, ∑ k = 1 ∞ ‍ 1 / λ k < ∞ andi n f k ( λ k + 1 - λ k ) > 0 . We investigate the Müntz spacesM p Λ = s p a n ¯ { t λ k : k = 1, 2, … } ⊂ L p ( 0, 1 ) for1 ≤ p ≤ ∞ . We show that, for eachp, there is a Müntz spaceF p which contains isomorphic copies of all Müntz spaces as complemented subspaces.F p is uniquely determined up to isomorphisms by this maximality property. We discuss explicit descriptions ofF p . In particularF p is isomorphic to a Müntz spaceM p ( Λ ^ ) whereΛ ^ consists of positive integers. Finally we show that the Banach spaces( ∑ n ‍ ⊕ F n ) p for1 ≤ p < ∞ and( ∑ n ‍ ⊕ F n ) 0 forp = ∞ are always isomorphic to suitable Müntz spacesM p ( Λ ) if theF n are the spans of arbitrary finitely many monomials over[ 0, 1 ] .