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Norm attaining polynomials
Author(s) -
Pappas A.,
Sarantopoulos Y.,
Tonge A.
Publication year - 2007
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdl033
Subject(s) - mathematics , norm (philosophy) , unit sphere , banach space , converse , hilbert space , dual norm , combinatorics , pure mathematics , geometry , political science , law
If L ≠ 0 is a continuous symmetric n ‐linear form on a Banach space and L ^ is the associated continuous n ‐homogeneous polynomial, the ratio ∥ L ∥ / ∥ L | | always lies between 1 and n n / n !. At one extreme, if L is defined on Hilbert space, then∥ L ∥ / ∥ L ^ ∥ = 1 . If L attains norm on Hilbert space, then L ^ also attains norm; in this case, we give an explicit construction to provide a unit vector x 0 with∥ L ^ ∥ = | L ^ ( x 0 ) | = ∥ L | . At the other extreme, if∥ L ∥ / ∥ L ^ ∥ = n n / n ! and L attains norm, then L ^ attains norm. We prove that in general the converse is not true.