Each vector in the Hilbert space is known as a ket, and written as
where ψ is an arbitrary label for the ket.
Each ket |ψ> has a dual vector, called a bra, written as
A bra <ψ′| and a ket |ψ> may form an inner product, called a bra-ket, or simply bracket. This is written as
[Outer products]? are written as |ψ′><ψ|. One use of the outer product is to construct a [projection operator]?. Given a ket |ψ>, the projection operator onto the subspace spanned by |ψ> is
Two Hilbert spaces V and W may form a third space V × W by a tensor product. If |ψ> is a ket in V and |φ> is a ket in W, the tensor product of the two kets is a ket in V × W. This is written variously as