Almost norming functionals, Part 1

Let E be a real Banach space with the dual E^*. By the Hahn-Banach theorem, for every unit vector e\in E there exists a functional e^*\in E^* of unit norm such that e^*(e)=1. One says that e^* is a norming functional for e. In general, one cannot choose e^* so that it depends continuously on e. For example, the 2-dimensional space with \ell_1 norm does not allow such a continuous selection.

Fix \delta\in (0,1) and call a linear functional e^* almost norming for e if |e|=|e^*|=1 and e^*(e)\ge \delta. 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 S_E is paracompact.

The longer proof uses paracompactness to obtain a locally finite cover of S_E 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 S_E was already mentioned.
  • Consider the set-valued map e\mapsto A(e) where A(e) is the closure of the set \{|e^*|\le 1, e^*(e)>\delta\}. The set A(e) is nonempty, convex and closed for each e.
  • The map e\mapsto A(e) is lower semicontinuous; that is, for any e and any open set V\subset E^* that meets A(e) there exists a neighborhood U of e such that V meets A(e') for all e'\in U. Indeed, V contains a point x\in \{|e^*|\le 1, e^*(e)>\delta\}. It is easy to see that x\in A(e') as long as \|e'-e\|<\delta-e^*(e).

By Michael’s selection theorem there exists a continuous map f\colon S_E\to E^* such that f(e)\in A(e) for all e\in S_E. It remains to normalize it: \frac{f(e)}{\|f(e)\|} is also continuous. QED

The above argument does not give uniform continuity of f\colon S_E\to S_{E^*}. Indeed, there is no uniformly continuous selection of almost norming functionals in \ell_1. 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.

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