  In this paper, we present the necessary and sufficient conditions of
separability for bipartite pure states in infinite dimensional Hilbert spaces.
Let $M$ be the matrix of the amplitudes of $\ket\psi$, we prove $M$ is a
compact operator. We also prove $\ket\psi$ is separable if and only if $M$ is a
bounded linear operator with rank 1, that is the image of $M$ is a one
dimensional Hilbert space. So we have related the separability for bipartite
pure states in infinite dimensional Hilbert spaces to an important class of
bounded linear operators in Functional analysis which has many interesting
properties.
