  Contrary to the simple structure of the tensor product of the quaternionic
Hilbert space, the octonionic situation becomes more involved. It turns out
that an octonionic Hilbert space can be decomposed as an orthogonal direct sum
of two subspaces, each of them isomorphic to a tensor product of an irreducible
octonionic Hilbert space with a real Hilbert space. As an application, we find
that for a given orthogonal basis the octonionic Parseval equality holds if and
only if the basis is weak associative. Fortunately, there always exists a weak
associative orthogonal basis in an octonionic Hilbert space. This completely
removes the obstacles caused by the failure of the octonionic Parseval
equality. As another application, we provide a new approach to studythe
Cayley-Dickson algebras, which turn out to be specific examples of octonionic
Hilbert spaces. An explicit weak associative orthonormal basis is constructed
in each Cayley-Dickson algebra.
