Let
X and
Y be two normed spaces. Let
U be a non-principal ultrafilter on
N. Let
g:X→Y be a standard
ε-phase isometry for some
ε≥0, i.e.,
g(0)=0, and for all
u,v∈X,
||‖g(u)+g(v)‖±‖g(u)−g(v)‖|−|‖u+v‖±‖u−v‖||≤ε.
The mapping
g is said to be a phase isometry provided that
ε=0.
In this paper, we show the following universal inequality of
g: for each
u∗∈w∗-exp
‖u∗‖BX∗, there exist a phase function
σu∗:X→{−1,1} and
φ ∈ Y∗ with
‖φ‖=‖u∗‖≡α satisfying that
|⟨u∗,u⟩−σu∗(u)⟨φ,g(u)⟩|≤52εα,forallu∈X.
In particular, let
X be a smooth Banach space. Then we show the following:
(1) the universal inequality holds for all
u∗∈X∗;
(2) the constant
52 can be reduced to
32 provided that
Y∗ is strictly convex;
(3) the existence of such a
g implies the existence of a phase isometry
Θ:X→Y such that
Θ(u)=limn,Ug(nu)n provided that
Y∗∗ has the
w∗-Kadec-Klee property (for example,
Y is both reflexive and locally uniformly convex).
