 
Summary: arXiv:1005.1050v5[math.FA]31Dec2010
REAL ANALYTIC APPROXIMATION OF LIPSCHITZ
FUNCTIONS ON HILBERT SPACE AND OTHER BANACH
SPACES
D. AZAGRA, R. FRY, AND L. KEENER
Abstract. Let X be a separable Banach space with a separating poly
nomial. We show that there exists C 1 (depending only on X)
such that for every Lipschitz function f : X R, and every > 0,
there exists a Lipschitz, real analytic function g : X R such that
f(x)  g(x) and Lip(g) CLip(f). This result is new even in the
case when X is a Hilbert space. Furthermore, in the Hilbertian case we
also show that C can be assumed to be any number greater than 1.
1. Introduction and main results
Not much is known about the natural question of approximating func
tions by real analytic functions on a real Banach space X. When X is
finite dimensional, a famous paper of Whitney's [W] provides a completely
satisfactory answer to this problem: a combination of integral convolutions
with Gaussian kernels and real analytic approximations of partitions of unity
allows to show that for every Ck function f : Rn Rm and every contin
uous : Rn (0, +) there exists a real analytic function g such that
