scalar product map $:\bra{\phi}\otimes\ket{\psi}\mapsto\braket{\phi}
{\psi}$. All of these usages are encompassed by the same spin network. The identity spin network can also be seen as an element of $\CS_j\otimes\CS^*_j$, that is, the space dual to $\CS_j^*\otimes\CS_j$ on which it, viewed as a $\Complexes$-valued linear operator, acts (the third line of Fig. \[idj\]).

In general a spin network has some number of edges that terminate in incoming or outgoing chevrons. Examples are the identity diagram in Fig. \[roi\] and the Wigner intertwiner in Fig. \[Wintertwiner\]. In all cases there are multiple interpretations of the spin network as a linear operator mapping one vector space to another, depending on how many of the incoming and outgoing chevrons have ket and bra chevrons plugged into them (specifying the domain), and how many are left free (specifying the range). The domain is the tensor product of some number of $\CS_j$ times some number of $\CS_j^*$, and so is the range. In the extreme case that all incoming and outgoing chevrons on the spin network have kets and bras plugged into them the result is simply a number and the range is $\Complexes$. In that case the spin network, as a $\Complexes$-valued linear operator, can 