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Banach spaces with the $ 2$-summing property

Authors: A. Arias, T. Figiel, W. B. Johnson and G. Schechtman
Journal: Trans. Amer. Math. Soc. 347 (1995), 3835-3857
MSC: Primary 46B20
MathSciNet review: 1303114
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Abstract: A Banach space $ X$ has the $ 2$-summing property if the norm of every linear operator from $ X$ to a Hilbert space is equal to the $ 2$-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent of the scalar field: the property is self-dual and any space with the property is a finite dimensional space of maximal distance to the Hilbert space of the same dimension. In the case of real scalars only the real line and real $ \ell _\infty ^2$ have the $ 2$-summing property. In the complex case there are more examples; e.g., all subspaces of complex $ \ell _\infty ^3$ and their duals.

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