tailieunhanh - Simultaneous proximinality of vector valued function spaces

A characterization of best simultaneous approximation of K¨othe spaces of vector-valued functions is given. This characterization is a generalization of some analogous theorems for Orlicz Bochner spaces. | Turk J Math 36 (2012) , 437 – 444. ¨ ITAK ˙ c TUB doi: Simultaneous proximinality of vector valued function spaces Mona Khandaqji, Fadi Awawdeh, Jamila Jawdat Abstract A characterization of best simultaneous approximation of K¨ othe spaces of vector-valued functions is given. This characterization is a generalization of some analogous theorems for Orlicz Bochner spaces. Key words and phrases: Simultaneous approximation; K¨ othe Bochner function space 1. Introduction Through this paper, let (T, , μ) be a finite complete measure space and L0 = L0 (T ) denote the space of all (equivalence classes) of Σ-measurable real valued functions. For f, g ∈ L0 , f ≤ g means that f (t) ≤ g (t) μ-almost every where t ∈ T . A Banach space (E, · E ) is said to be a K¨ othe space if (1) for f, g ∈ L0 , |f| ≤ |g| and g ∈ E imply f ∈ E and f E ≤ g E ; (2) for each A ∈ Σ, if μ (A) is finite then χA ∈ E . See [7, p. 28]. A K¨ othe space E has absolutely continuous norm if for each f ∈ E and each decreasing sequence (An ) converges to 0 , then χAn f E → 0 . A K¨ othe space E is said to be strictly monotone if x ≥ y ≥ 0 and x E = y E imply x = y . Let E be a K¨ othe space on the measure space (T, , μ) and (X, · X ) be a real Banach space then E (X) is the space (of all equivalence classes) of strongly measurable functions f : T → X such that f (·) X ∈ E equipped with the norm | f | = f (·) X E . The space (E (X) , |·| E ) is a Banach space called the K¨othe Bochner function space [7]. For a function n F = (f1 , f2 , . . . , fn ) ∈ (E (X)) , we define the norm of F by n fi (·) X | F | = i=1 . E 2000 AMS Mathematics Subject Classification: 41A50, 41A28, 46E40. 437 KHANDAQJI, AWAWDEH, JAWDAT The most important classes of K¨othe Bochner function spaces are the Lebesgue Bochner spaces Lp (X) , (1 ≤ p 0 . Since E (X) is a K¨ othe Bochner function space with absolutely continuous norm, the simple functions are dense in E(X),

• Toán học
• Simultaneous proximinality
• Vector valued function spaces
• Simultaneous approximation
• Kothe Bochner function space
• Orlicz Bochner spaces