We start with a Hilbert space H. Each vector in H is known as a ket, and written as
where ψ is an arbitrary label for the ket. Each element of the dual space of H (i.e. each continuous linear function from H to the complex numbers C) is known as a bra, and written as
where φ is an arbitrary label for the bra. Applying the bra <φ| to the ket |ψ> results in a complex number, called a bra-ket, which we write as
Every ket |ψ> has a dual bra, written as <ψ|, a continuous linear function on H defined as follows:
for all bras |x>, where the right hand side ( , ) denotes the inner product given on the Hilbert space. The notations is justified by the [Riesz representation theorem]?, which states that every bra in the dual space arises from one and only one ket in this fashion.
[Outer products]? are written as |φ><ψ|. One use of the outer product is to construct [projection operators]?. Given a ket |ψ> of norm 1, 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