Ruang Lipschitz
Keywords:
Lipschitz Function, Scalar Valued Lipschitz Function, Lipschitz Space, Banach Space, Lipschitz NormAbstract
Given a metric space (X, ρ) and a field F (real or complex). A function f : X → F is said to be scalar-valued Lipschitz function if there exists a constant α ≥ 0 such that ρ(f(p), f(q)) ≤ α ρ(p, q), for all p, q ∈ X. Lipschitz space Lip(X) is the space of all bounded scalar valued Lipschitz function on X. Addition and scalar multiplication defined on Lip(X) with (f + g)(p) = f(p) + g(p) and (cf)(p) = c f(p), for all p ∈ X, f, g ∈ Lip(X), c ∈ F. Lipschitz space equipped with norm Lipschitz which is defined by ||f|| = max{||f||∞, L(f)}. This study observes the properties of scalar valued Lipschitz function and its relationship with Banach Space.
ABSTRAK
Diberikan ruang metrik (X, ρ) dan lapangan F (real atau kompleks). Suatu fungsi f : X → F dikatakan fungsi Lipschitz bernilai skalar jika terdapat konstanta α ≥ 0 sedemikian sehingga ρ(f(p), f(q)) ≤ α ρ(p, q), untuk setiap p, q ∈ X. Ruang Lipschitz Lip(X) adalah ruang dari semua fungsi Lipschitz terbatas bernilai skalar pada X. Didefinisikan penjumlahan dan perkalian skalar pada Lip(X) dengan aturan (f + g)(p) = f(p) + g(p) dan (cf)(p) = c f(p), untuk setiap p ∈ X, f, g ∈ Lip(X), c ∈ F. Ruang Lipschitz dilengkapi dengan norm Lipschitz yang didefinisikan sebagai ||f|| = max{||f||∞, L(f)}. Kajian ini mengkaji sifat-sifat dari fungsi Lipschitz bernilai skalar dan hubungannya dengan ruang Banach.
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