%0 Journal Article
%A Bombal Gordón, Fernando
%A Fernández Unzueta, M.
%T Polynomial convergence of sequences in Banach spaces.
%D 2000
%@ 1137-2141
%U https://hdl.handle.net/20.500.14352/58310
%X Let E  be a (complex) Banach space and n  be a positive integer, and denote by P( n E)  the space of all n -homogeneous polynomials on E . The authors say that E  has the m -FJ property (m -Farmer-Johnson property) if, whenever (x n ) n   is a sequence in E  which converges weakly to x  in E  and P(x n )  converges to P(x)  for all P∈P( m E) , then Q(x n )  converges to Q(x)  for all Q∈P( k E)  for all k , 1≤k≤m . If all m -homogeneous polynomials are weakly sequentially continuous, then E  has the m -FJ property. Using this observation, the authors give an alternative proof of the result that if E  is a Banach space such that every m -homogeneous polynomial on E  is weakly continuous on bounded sets then every k -homogeneous polynomial, 1≤k≤m , is weakly continuous on bounded sets [C. Boyd and R. A. Ryan, Arch. Math. (Basel) 71 (1998), no. 3, 211–218;]. It is shown that the m -FJ property is equivalent to the condition that, whenever y∈E  and (x n ) n   is a sequence in E  which converges weakly to x  in E  and moreover P(x n )  converges to P(x)  for all P∈P( m E) , then P(x n +y)  converges to P(x+y)  for all P∈P( m E) .    The main result of the paper is the following: Let E  be a Banach space with an unconditional decomposition E=∑ ∞ k=1 E k  . Suppose each E k   is a Banach space such that, if (x n ) n   is a sequence in E k   with the property that whenever x n   converges weakly to x  and P(x n )  converges to P(x)  for all P∈P( m E k ) , then x k   converges in norm to x . Then E  has the m -FJ property. If J  denotes the James space, a corollary to the main theorem is that the space (∑⊗J) l p   , 1≤p<∞ , has the m -FJ property for every integer m .
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