范数在平面上以11范数得到双线性形式

Pub Date : 2022-11-01 DOI:10.2478/ausm-2022-0008

Sung Guen Kim

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引用次数: 0

摘要

抽象为给定的单位向量x1,···,xn真正的巴拿赫空间E,我们定义NA(ℒ(nE)) (x1,……xn) = {T∈ℒ(nE): | T (x1,…xn) | =为T为= 1},NA \离开({\ mathcal {L} \离开({^ nE} \右)}\)\离开({{x_1}, \ ldots {x_n}} \右)左= \ \ {{T \ \ mathcal {L} \离开({^ nE} \右):\左| {T \离开({{x_1}, \ ldots {x_n}} \右)}\右左| = \ \ | T \ \ | = 1} \右\},在ℒ(nE)表示所有连续n-linear形式的巴拿赫空间E具有常态| | T | | =一口| | xk | | = 1, 1≤k≤n | T (x1,。,xn) |。本文对单位向量(x1, x2)， (y1, y2)∈l12，其中l12 =具有l1范数的l2，我们对NA(f (2l12))((x1, x2)， (y1, y2))进行分类。

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Norm attaining bilinear forms on the plane with the l1-norm

Abstract For given unit vectors x1, · · ·, xn of a real Banach space E, we define NA(ℒ(nE))(x1,…xn)={ T∈ℒ(nE):| T(x1,…xn) |=‖ T ‖=1 }, NA\left( {\mathcal{L}\left( {^nE} \right)} \right)\left( {{x_1}, \ldots {x_n}} \right) = \left\{ {T \in \mathcal{L}\left( {^nE} \right):\left| {T\left( {{x_1}, \ldots {x_n}} \right)} \right| = \left\| T \right\| = 1} \right\}, where ℒ(nE) denotes the Banach space of all continuous n-linear forms on E endowed with the norm ||T|| = sup||xk||=1,1≤k≤n |T(x1, . . ., xn)|. In this paper, we classify NA(ℒ(2l12))((x1, x2), (y1, y2)) for unit vectors (x1, x2), (y1, y2)∈ l12, where l12 = ℝ2 with the l1-norm.

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