Let S be a closed convex subset of a Euclidean space X and T be another set containing S. There exists a continuous function g: T → S that retracts T onto S, that is, for which g(x) = x for every x ∊ S.
Earlier (exercise 3.64) we showed that every element in an inner product space defines a distinct continuous linear functional on the space. We now show that for a complete linear space, every continuous linear functional takes this form.