یکنواخت جداسازی در فضای توابع لیپ‌شیتس کوچک برداری مقدار

فرض کنید (x,d) یک فضای متریک فشرده و e یک فضای باناخ است. در این مقاله خاصیت یکنواخت جداسازی نقاط  توسط فضای lip0(x,e) مورد مطالعه قرار می‌گیرد.

uniformly separation property in vector-valued littlelipschitz space

suppose that (𝑋, 𝑑) be a compact metric space with a distinguished point𝑒 and𝐸 be a banach space.collection of 𝐸 −valued function 𝑓 on 𝑋 such thatℒ(𝑓) = sup,𝑥≠𝑦𝑥,𝑦∈𝑋‖𝑓(𝑥) − 𝑓(𝑦)‖𝑑(𝑥, 𝑦)< ∞ , 𝑓(𝑒) = 0is called vector-valued lipschitz space and denoted by 𝐿𝑖𝑝0 (𝑋, 𝐸). the space𝐿𝑖𝑝0(𝑋, 𝐸) with respect to the point wise operations on functions and the normℒ(. ) is a banach space that separates points of 𝑋. the subset consists of allfunctions such thatlim𝑑(𝑥,𝑦)→0‖𝑓(𝑥) − 𝑓(𝑦)‖𝑑(𝑥, 𝑦)= 0is a closed subspace of 𝐿𝑖𝑝0 (𝑋, 𝐸), denoted by 𝑙𝑖𝑝0(𝑋, 𝐸) and called littlevector-valued lipschitz space. in particular when banach space 𝐸 coincideswith scaler field, 𝐿𝑖𝑝0(𝑋, 𝐸) and 𝑙𝑖𝑝0(𝑋, 𝐸) is denoted by 𝐿𝑖𝑝0(𝑋) and 𝑙𝑖𝑝0(𝑋)respectively.definition. the space 𝑙𝑖𝑝0(𝑋) separates points of 𝑋 uniformly when there exists𝐶 > 1 such that for each distinct pair point 𝑥, 𝑦 ∈ 𝑋 there is 𝑓 ∈ 𝑙𝑖𝑝0(𝑋, 𝐸) with𝑓(𝑦) = 0, ‖𝑓(𝑥)‖ = 𝑑(𝑥, 𝑦), ℒ(𝑓) ≤ 𝐶.definition.the banach space 𝐸 has approximation property if for each ε > 0and compact subset 𝐾 of 𝐸 there exists a finite dimensional bounded operator𝑇: 𝐸 → 𝐸 such that sup𝑥∈𝐾‖𝑇𝑥 − 𝑥‖ < 𝜀.results and discussionin this paper we deal with the uniform separation property of a metric space 𝑋by the little vector-valued lipschitz space, namely 𝑙𝑖𝑝0(𝑋, 𝐸).conclusionwe show that if 𝑙𝑖𝑝0 (𝑋) has the approximation property and 𝐸 be a topologicaldual of some banach space, then there exists a compact metric space 𝑌 with adistinguished point and a non-expansive function 𝜋: 𝑋 → 𝑌 such that 𝑙𝑖𝑝0(𝑌, 𝐸)separates the point of 𝑌 uniformly and 𝐶𝜋, the composition operator induced by𝜋, is a surjective linear isometry from 𝑙𝑖𝑝0(𝑌, 𝐸) to 𝑙𝑖𝑝0(𝑋, 𝐸).