Let be a real Banach space with the dual . By the Hahn-Banach theorem, for every unit vector there exists a functional of unit norm such that . One says that is a norming functional for . In general, one cannot choose so that it depends continuously on . For example, the 2-dimensional space with norm does not allow such a continuous selection.
Fix and call a linear functional almost norming for if and . In any Banach space there exists a continuous selection of almost norming functionals.
I know two proofs, neither of which is hard. Both rely on Stone’s theorem: every metric space is paracompact. In particular, the unit sphere is paracompact.
The longer proof uses paracompactness to obtain a locally finite cover of by sets contains in thin spherical caps.
The shorter proof uses Michael’s continuous selection theorem. So we must verify its hypotheses.
- The paracompactness of was already mentioned.
- Consider the set-valued map where is the closure of the set . The set is nonempty, convex and closed for each .
- The map is lower semicontinuous; that is, for any and any open set that meets there exists a neighborhood of such that meets for all . Indeed, contains a point . It is easy to see that as long as .
By Michael’s selection theorem there exists a continuous map such that for all . It remains to normalize it: is also continuous. QED
The above argument does not give uniform continuity of . Indeed, there is no uniformly continuous selection of almost norming functionals in . It is not clear exactly which spaces admit uniformly continuous selection: superreflexivity is sufficient, but is it necessary? Same question can be posed for Holder continuous or Lipschitz continuous selections.