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.