  The study of Gaussian measures on Banach spaces is of active interest both in
pure and applied mathematics. In particular, the spectral theorem for
self-adjoint compact operators on Hilbert spaces provides a canonical
decomposition of Gaussian measures on Hilbert spaces, the so-called
Karhunen-L{\`o} eve expansion. In this paper, we extend this result to Gaussian
measures on Banach spaces in a very similar and constructive manner. In some
sense, this can also be seen as a generalization of the spectral theorem for
covariance operators associated to Gaussian measures on Banach spaces. In the
special case of the standard Wiener measure, this decomposition matches with
Paul L{\'e}vy's construction of Brownian motion.
