1997, VOLUME 3, NUMBER 1, PAGES 37-45

Polynomial continuity

José G. Llavona


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A mapping f: X ® Y between Banach spaces X and Y is said to be polynomially continuous (P-continuous, for short) if its restriction to any bounded set is uniformly continuous for the weak polynomial topology, i.e., for every e > 0 and bounded B Ì X, there are a finite set {p1, ¼ ,pn} of polynomials on X and d > 0 so that ||f(x)-f(y)|| < e whenever x,y Î B satisfy |pj(x-y)| < d (1 £ j £ n). Every compact (linear) operator is P-continuous. The spaces L¥[0,1], L1[0,1] and C[0,1], for example, admit polynomials which are not P-continuous.

We prove that every P-continuous operator is weakly compact and that for every k Î N (k ³ 2) there is a k-homogeneous scalar valued polynomial on $ \ell _1 $ which is not P-continuous.

We also characterize the spaces for which uniform continuity and P-continuity coincide, as those spaces admitting a separating polynomial. Other properties of P-continuous polynomials are investigated.

All articles are published in Russian.

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Last modified: November 16, 1999