The dimensional of a subset of lp as p varies

Abstract

We investigate how the dimension of a set X contained in l p can change as p is varied. Assume 1 ⩽ s < p < q < r < ∞, and X ⊆ l p ⊆ l q ⊆ l r . An example shows that dim p X can be greater than dim T X. A result states that if dim p X > dim r X, then dim p X = dim r X + 1, dim q X = dim r X, and X is not a subset of l s . Another example shows that it is possible that dim r X > dim p X. It is shown, for example, that the dimension of the rational points in l 2 is zero when this set is viewed as a subset of l p where p > 2. There is a positive dimensional closed subset of l p whose projection onto each coordinate axis is a two point set; this subset admits a natural topological group structure.