# 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.