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Extremal Homogeneous Polynomials on Real Normed Spaces

✍ Scribed by Pádraig Kirwan; Yannis Sarantopoulos; Andrew M Tonge

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
135 KB
Volume
97
Category
Article
ISSN
0021-9045

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✦ Synopsis

If P is a continuous m-homogeneous polynomial on a real normed space and P 8 is the associated symmetric m-linear form, the ratio &P 8 &Â&P& always lies between 1 and m m Âm!. We show that, as in the complex case investigated by Sarantopoulos (1987, Proc. Amer. Math. Soc. 99, 340 346), there are P 's for which &P 8 &Â&P&= m m Âm! and for which P 8 achieves norm if and only if the normed space contains an isometric copy of l m 1 . However, unlike the complex case, we find a plentiful supply of such polynomials provided m 4.

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