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Continuity of superposition operators on sequence spaces. (English) Zbl 0840.46001
Summary: Let $$E$$ be a sequence space and $$Y$$ a topological vector space. A map $$G: E\to Y$$ is said to be orthogonally additive if $$G(x+ y)= G(x)+ G(y)$$ whenever $$x$$ and $$y$$ have disjoint supports. We show that a certain class of orthogonally additive maps defined on a sequence space satisfying a gliding hump condition are automatically continuous. We also establish versions of the uniform boundedness and Banach-Steinhaus theorems for such operators.

##### MSC:
 46A32 Spaces of linear operators; topological tensor products; approximation properties 46A45 Sequence spaces (including Köthe sequence spaces) 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)