A well-known Stein's theorem asserts that, in order that a
real-valued random variable $mathbf{X}$ has a standard normal
distribution it is necessary and sufficient that, for all
continuous and piecewise continuously differentiable functions
$f:mathbb{R}
ightarrowmathbb{R}$ with finite $Emid
f'(mathbf{X})mid$, the following identity holds:
egin{eqnarray}label{a1}
E[f'(mathbf{X})]=E[mathbf{X}f(mathbf{X})].
end{eqnarray}

In this paper we concern ourself with the generalization of
Stein's characterization to a random variable $mathbf{X}$ taking
values in a real separable Banach space $B$. Our main result show
that, in order that the probability measure
$mu=Pcircmathbf{X}^{-1}$ of $mathbf{X}$ is standard Gaussian
it is necessary and sufficient that, there exists a real separable
Hilbert space $H$ such that $(i,H,B)$ is an abstract Wiener space
in the sense of Gross and the following identity holds:
egin{eqnarray}label{a2}
int_{B}(x,z)f((x,y))mu(dx) = l z,ygint_{B}f'((x,y))mu(dx),
end{eqnarray}
for $y, zin B^{*}$and for any differentiable function $f$ such that
$int_{B}|f'((x,y))|mu(dx)$ is finite, where $(cdot,cdot)$ and
$lcdot,cdotg$ denote respectively the $B-B^*$ pairing and the
inner product of $H$.

[1]R. M. Dudley, J. Feldman, L. LeCam,: On semi-norms and
probabilities, and Abstract Wiener spaces, Ann. , Math.
93(1971), 390-408.

[2]L. Gross, Abstract Wiener space,
Proc. 5th. Berkeley Sym. Math. Stat. Prob. 2, part 1
(1965) 31-42, Universoty of California Press, Berkeley.

[3]K. Erwin,
Introductory functional analysis with applications,
Wiley, (1978).

[4]H. H. Kuo, Gaussian Measure in Banach Space, Lecture Notes in Math. 463, Spring-Verlag (1975).

[5]H. H. Kuo, White Noise Distribution Theory, Probability and Stochastics Series, CRC Press, (1996).

[6]H. H. Kuo, Integration by parts for abstract Wiener measures, Duke Math. J. 41(1974), 373-379.

[7]Y. J. Lee, Applications of the Fourier-Wiener transform to differential equations on infinite dimensional spaces I, Trans. Amer. Soc., 262(1980), 259-283.

[8]C. Stein, Approximate computation of expectations, Lecture notes-monograph series ; vol. 7, (1986) Institute of Mathematical Statistics, Hayward, California.