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.
