Smooth norms in dense subspaces of Banach spaces

Abstract

In the first part of our paper, we show that ℓ ∞ has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as ℓ 1 (c), also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of c 0 (ω 1 ) admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler in [8]) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a C ∞ -smooth norm. Finally, we prove that there is a dense subspace Y of ℓ c ∞ (ω 1 ) such that every renorming of Y contains an isometric copy of c 00 (ω 1 ).

Smooth and polyhedral norms via fundamental biorthogonal systems

“…Vanderwerff [58] proved that every normed space with a countable algebraic basis admits a C 1 -smooth norm; this result was later improved to obtain a C ∞ -smooth norm [26], a polyhedral norm [13], and an analytic one [13]. These results and the previous discussion motivated [24, Problem 149], [32], and a recent research of the present authors [9], where the following problem was posed.…”

“…Here we should observe that, for a normed space, admitting a countable algebraic basis is equivalent to being the linear span of the vectors of an M-basis, again by [40]. On the other hand, in [13] an analytic norm is also constructed in normed spaces with a countable algebraic basis, while in our result it is not possible in general to obtain analytic norms, [9,Theorem 3.10].…”

“…Although the problem is seemingly very general and ambitious, note that [13] answers it in the positive for X separable and k = ω. Moreover, in [9] it was solved in the positive for ℓ ∞ and k = ω, ℓ 1 (c) and k = ω, and spaces with long unconditional bases and k = ∞. It is worth pointing out that in these results the density of smooth norms cannot be guaranteed in general.…”

“…For non-separable Banach spaces the problem has only been faced in [9,Theorem B], where a C ∞ -smooth norm is constructed in the linear span of every long unconditional Schauder basis (and in [9, Theorem A], concerning the concrete spaces ℓ ∞ and ℓ 1 (c), as mentioned above). Once more, Theorem A is substantially stronger, since we additionally obtain an approximation result, the LFC condition, polyhedral norms, partitions of the unity, and C 1 -smooth LUR norms.…”

On densely isomorphic normed spaces

“…It was shown in [20] that every separable Banach space admits a dense subspace with C ∞ -smooth renorming-in sharp contrast with the space itself. In our forthcoming joint paper with Sheldon Dantas [7] we prove similar results for certain non-separable Banach spaces, e.g., those having an unconditional basis. However, our proofs rely on the structural properties of the selected subspaces, and do not work for a general dense subspace.…”