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Lower bounds for multilinear forms defined on Hilbert Spaces
Author(s) -
García-Vázquez Juan Carlos,
Villa Rafael
Publication year - 1999
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300007774
Subject(s) - multilinear map , mathematics , hilbert space , unit (ring theory) , pure mathematics , hilbert r tree , space (punctuation) , mathematical analysis , discrete mathematics , reproducing kernel hilbert space , rigged hilbert space , linguistics , philosophy , mathematics education
In this paper, it is proved that, for any m unit vectors. x 1 …, x m in any n ‐dimensional real Hilbert space, there exists a unit vector x 0 such that| 〈 x 1 , x 0 〉 ⋯ 〈 x m , x 0 〉 |1 / m = exp (∫ S n − 1log ( | 〈 x , y 〉 | ) d σ n − 1( x ))for any y ∈ S n −1 . The exact value of the above integral is calculated, and these results used to improve some lower bounds for multilinear forms on real Hilbert spaces. An integral expression is also given for the complex case.