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Extremal multilinear forms on Banach spaces

Author: I. Sarantopoulos
Journal: Proc. Amer. Math. Soc. 99 (1987), 340-346
MSC: Primary 46B20; Secondary 46G20
MathSciNet review: 870797
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Abstract: Suppose that $ L$ is a continuous symmetric $ m$-linear form defined on a complex Banach space $ E$, and $ \hat L$ is the associated homogeneous polynomial. If

$\displaystyle \vert\vert L \vert\vert = ({m^m}/m!)\vert\vert {\hat L} \vert\vert,$

we prove that $ E$ contains an almost isometric copy of $ l_m^1$. In particular if $ E$ is an $ m$-dimensional space, then $ E$ is isometrically isomorphic to $ l_m^1$. We also prove that the only examples of such extremal $ L$ which achieve their norm are suitable "extensions" of a known example given by Nachbin.

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