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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.

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.)