  We present an alternative construction of the infinite dimensional It\^{o}
integral with respect to a Hilbert space valued L\'{e}vy process. This approach
is based on the well-known theory of real-valued stochastic integration, and
the respective It\^{o} integral is given by a series of It\^{o} integrals with
respect to standard L\'{e}vy processes. We also prove that this stochastic
integral coincides with the It\^{o} integral that has been developed in the
literature.
