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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Bounds for projection constants and $ 1$-summing norms

Authors: Hermann König and Nicole Tomczak-Jaegermann
Journal: Trans. Amer. Math. Soc. 320 (1990), 799-823
MSC: Primary 46B10; Secondary 47B10
MathSciNet review: 968885
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Abstract: It is shown that projection constants $ \lambda ({X_n})$ of $ n$-dimensional normed spaces $ {X_n}$ satisfy $ \lambda ({X_n}) \leqslant \sqrt n - c/\sqrt n $ where $ c > 0$ is a numerical constant. Similarly, the $ 1$-summing norms of (the identity of) $ {X_n}$ can be estimated by $ {\pi _1}({X_n}) \geqslant \sqrt n + c/\sqrt n $. These estimates are the best possible: for prime $ n$, translation-invariant $ n$-dimensional spaces $ {X_n}$ such that $ \lambda ({X_n}) \geqslant \sqrt n - 2/\sqrt n $ and $ {\pi _1}({X_n}) \leqslant \sqrt n + 2/\sqrt n $ can be constructed. For these spaces Gordon-Lewis constants and distances to Hilbert spaces are large as well: $ \operatorname{gl} ({X_n}) \geqslant \tfrac{1} {3}\sqrt n ,d({X_n},l_2^n) = \sqrt n $.

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