The dimensional of a subset of lp as p varies

AbstractWe investigate how the dimension of a set X contained in lp can change as p is varied.Assume 1 ⩽ s < p < q < r < ∞, and X ⊆ lp ⊆ lq ⊆ lr. An example shows that dimp X can be greater than dimT X. A result states that if dimp X > dimr X, then dimp X = dimr X + 1, dimq X = dimr X, and X is not a subset of ls. Another example shows that it is possible that dimr X > dimp X.It is shown, for example, that the dimension of the rational points in l2 is zero when this set is viewed as a subset of lp where p > 2. There is a positive dimensional closed subset of lp whose projection onto each coordinate axis is a two point set; this subset admits a natural topological group structure