NORM ATTAINING MULTILINEAR FORMS ON Rn WITH THE ℓ1-NORM

초록

Let n, m ∈ N with n, m ≥ 2. For given unit vectors x<inf>1</inf>, · · ·, x<inf>n</inf> of a real Banach space E, we define NA(L(n E))(x<inf>1</inf>, · · ·, x<inf>n</inf>) = {T ∈ L(n E): |T (x<inf>1</inf>, · · ·, x<inf>n</inf>)| = ∥T ∥ = 1}, where L(n E) denotes the Banach space of all continuous n-linear forms on E endowed with the norm ∥T ∥ = sup<inf>∥xk</inf> ∥=1,1≤k≤n |T (x<inf>1</inf>, …, x<inf>n</inf>)|. In this paper, we present a characterization of the elements in the set NA(L(m ℓn1))(W<inf>1</inf>, · · ·, W<inf>m</inf>) for any given unit vectors W<inf>1</inf>, …, W<inf>m</inf> ∈ ℓn1, where ℓn1 = Rn with the ℓ<inf>1</inf>-norm. This result generalizes the results from [7], and two particular cases for it are presented in full detail: the case n = 2, m = 2, and the case n = 3, m = 2. © 2024, Transilvania University of Brasov 1. All rights reserved.

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