(Tensor product splitting) Let
and
be Hilbert spaces and
and
are stable splittings for
and
respectively with condition numbers
and
.
Then
is a stable splitting of
with condition number
.

Proof

According to the section (
Stable
space splittings
) it suffices to derive the Schwarz operator for the
collection
and show that it is symmetric, positive and has strictly positive bounds. Such
derivation is a straightforward exercise.