Extreme points and isometries on vector-valued Lipschitz spaces

作者：Botelho Fernanda; Fleming Richard J^*; Jamison James E

来源：Journal of Mathematical Analysis and Applications, 2011, 381(2): 821-832.

DOI：10.1016/j.jmaa.2011.03.062

• Extreme points
• Isometry
• Lipschitz function

For a Banach space E and a compact metric space (X, d), a function F : X -> E is a Lipschitz function if there exists k > 0 such that
parallel to F(x) - F(y)parallel to <= kd(x,y) for all x,y is an element of X.
The smallest such k is called the Lipschitz constant L(F) for F. The space Lip(X, E) of all Lipschitz functions from X to E is a Banach space under the norm defined by
parallel to F parallel to = max{L(F), parallel to F parallel to(infinity)},
where parallel to F parallel to(infinity) = sup{parallel to F(x)parallel to: x is an element of X}. Recent results characterizing isometrics on these vector-valued Lipschitz spaces require the Banach space E to be strictly convex. We investigate the nature of the extreme points of the dual ball for Lip(X, E) and use the information to describe the surjective isometries on Lip(X, E) under certain conditions on E. where E is not assumed to be strictly convex. We make use of an embedding of Lip(X, E) into a space of continuous vector-valued functions on a certain compact set.

• 出版日期2011-9-15