Abstract
Sharp triangle inequality and its reverse in Banach spaces were recently showed by Mitani et al. (2007). In this paper, we present equality attainedness for these inequalities in strictly convex Banach spaces.
Highlights
The triangle inequality and its reverse inequality have been treated in 1–5 see 6, 7 .Kato et al 8 presented the following sharp triangle inequality and its reverse inequality with n elements in a Banach space X
We present equality attainedness for 1.4 and 1.5 in Theorem 1.2
We will prove Theorem 3.7 by the induction. Assume that this theorem holds true for all nonzero elements in X less than n
Summary
The triangle inequality and its reverse inequality have been treated in 1–5 see 6, 7 .Kato et al 8 presented the following sharp triangle inequality and its reverse inequality with n elements in a Banach space X. Presented these inequalities for strongly integrable functions with values in a Banach space. In this paper we first present a simpler proof of Theorem 1.2. To do this we consider the case x1 > x2 > · · · > xn , as follows. We consider equality attainedness for sharp triangle inequality and its reverse inequality in strictly convex Banach spaces. According to Theorem 1.1 Inequalities 1.4 and 1.5 hold for the case n 2 cf 3. For all positive numbers m with m > n let xk,m k m xk , k. As m → ∞, we have Inequalities 1.4 and 1.5
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