Some remarks on smooth renormings of Banach spaces

Abstract

We prove that in every separable Banach space $X$ with a Schauder basis and a $C^k$-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a $C^k$-smooth one in a way that the approximation is improving as fast as we wish on the elements depending only on the tail of the Schauder basis. Our result solves a problem from the recent monograph of Guirao, Montesinos and Zizler.

13 pp

References in corpus (4)

• Smooth and polyhedral approximation in Banach spaces
• Smooth norms and approximation in Banach spaces of the type C(K)
• Using boundaries to find smooth norms
• $C^k$-smooth approximations of LUR norms

Cited by in corpus (1)

• Smooth and polyhedral norms via fundamental biorthogonal systems