  Current work defines Schmidt representation of a bilinear operator $T: H_1
\times H_2 \rightarrow K$, where $H_1, H_2$ and $K$ are separable Hilbert
spaces. Introducing the concept of singular value and ordered singular value,
we prove that if $T$ is compact, and its singular values are ordered, then $T$
has a Schmidt representation on real Hilbert spaces. We prove that the
hypothesis of existence of ordered singular values is fundamental.
