Bases in spaces of homogeneous polynomials on Banach spaces

Leonardo Pellegrini
2007 Journal of Mathematical Analysis and Applications  
In this work we present some conditions of equivalence for the existence of a monomial basis in spaces of homogeneous polynomials on Banach spaces. basis of c 0 is a polynomial-shrinking basis. If E = l 1 , the canonical basis is not a polynomialshrinking basis, since it is not a shrinking basis. (b) Let E = d * (w, 1) be the predual of Lorentz's sequences space. In [9], Payá and Sevilla have proved that P w ( m d * (w, 1)) = P ( m d * (w, 1) ) if, and only if, w / ∈ l m , where l m is the
more » ... of the m-summable sequences. Then, because of the item (iv), the canonical basis (shrinking) of d * (w, 1) is a polynomial shrinking basis if, and only if, w / ∈ l p , ∀p > 1. (c) Let E = T * J be the Tsirelson-James' space. T * J has a shrinking basis and P w ( m T * J ) = P ( m T * J ) (see [6, p. 124]). So, T * J has a polynomial-shrinking basis. Application
doi:10.1016/j.jmaa.2006.09.058 fatcat:j53bp5byxjenvffhfgcrybcwju