be seen as a vector in the space dual to the domain.

This facile identification of closely associated operators, and their reinterpretation as elements of vector spaces, is an important advantage of spin networks. It is difficult and awkward to do something similar with the Dirac notation.

In general, a tensor is a multilinear operator acting on a tensor product of some set of vector spaces and their duals. Thus, a spin network is a notation for a tensor on some product of the $\CS_j$ and their duals. The edges of the spin network terminating in ket or bra chevrons indicate the nature of the space the tensor acts on. In general, the tensor product of two tensors in spin network notation is indicated by the placing of the two spin networks together on the page, in any orientation. The outer product of a bra and a ket illustrated in Fig. \[roi\] is a special case. Partial or complete contractions of tensors are indicated by joining some or all ket chevrons with bra chevrons.

The $2j$ symbol and intertwiner
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We now consider another intertwiner, which leads to an important mapping between kets and bras, alternative to Hermitian conjugation. The 