first, into the bra chevrons of the intertwiner, as in the final diagram of Fig. \[3jnetwork\].

More generally, a $\Complexes$-valued intertwiner on a Hilbert space $\HS$ can be regarded as an $SU(2)$-invariant bra vector on this space, that is, a member of $\HS^*$. By Hermitian conjugation we obtain an $SU(2)$-invariant ket vector in $\HS$. Thus, there is a one-to-one correspondence between the subspace of $\HS$ of rotationally invariant vectors and the set of $\Complexes$-valued intertwiners on $\HS$. The subspace $\ZS$ introduced in Sec. \[4jmodel\] below is a subspace of this type, consisting of rotationally invariant vectors.

Tensor products and resolution of identity {#tensorprodsroi}
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The outer product of a ket with a bra is represented in spin network language simply by placing the spin networks for the ket and the bra on the same page, as illustrated in Fig. \[roi\]. The orientations of the bra and ket spin networks is immaterial, but in the figure they have been placed with their $m$ indices adjacent in order to emphasize the summation (a contraction over two indices, one covariant, one contravariant). The final diagram is the spin network for the identity operator on space $\CS_j$.

Figure \[roi\] illustrates another technique, the replacement 