An indecomposable Banach space of continuous functions which has small density

A Hereditarily Indecomposable Asymptotic `2 Banach Space

A famous open problem in functional analysis is whether there exists a Banach space X such that every (bounded linear) operator on X has the form λ+K where λ is a scalar and K denotes a compact operator. This problem is usually called the “scalar-plus-compact” problem [14]. One of the reasons this problem has become so attractive is that by a result of N. Aronszajn and K.T. Smith [7], if a Bana...

The Space of Density Continuous Functions

A function f : R → R is density continuous if it is continuous when using the density topology on both its domain and range. The set of density continuous functions contains all convex functions and all real analytic functions, but does not contain all C∞ functions. It is not closed under addition. We denote by Rd the set of real numbers, R, endowed with the density topology. A function f : Rd ...

Spaces of continuous functions into a Banach space

An Exotic Minimal Banach Space of Functions

This note describes a new Banach space B0 of square integrable functions on R having many interesting invariance properties. In fact, the Fourier transform, time-frequency shifts, and L-normalized dilations act isometrically on it. For its definition, we make use of a general construction principle for minimal invariant spaces. We demonstrate a variety of properties following immediately from t...

Interpolating Hereditarily Indecomposable Banach Spaces

A Banach space X is said to be Hereditarily Indecomposable (H.I.) if for any pair of closed subspaces Y , Z of X with Y ∩ Z = {0}, Y + Z is not a closed subspace. (Throughout this section by the term “subspace” we mean a closed infinite-dimensional subspace of X .) The H.I. spaces form a new and, as we believe, fundamental class of Banach spaces. The celebrated example of a Banach space with no...