ket or bra chevrons.

To convert the ket chevron in Fig. \[Xket\] into a bra chevron we simply insert it into the second operand of $K:\CS_j\otimes\CS_j\to\Complexes$, as shown in Fig. \[lowerindex\]. The remainder of the spin network, indicated by the $X$, does not change.

Bras to kets {#brastokets}
------------

The map $K_1:\CS_j\to\CS_j^*$ has an inverse, the map $K_1^{-1}
:\CS_j^*\to\CS_j$, that takes bras into kets. We associate $K_1^{-1}$ with a closely related map $K^{-1}:\CS_j^*\otimes\CS_j^*\to\Complexes$ that is expressed by the spin network in Fig. \[Kinv\]. This spin network and the meaning of the $-1$ on $K^{-1}$ are to be defined, but the spin network represents a linear operator $:\CS_j^
*\otimes\CS_j^*\to\Complexes$ with the ordering of the two operands being specified by the stub. The action of $K^{-1}$ on two bras is illustrated in Fig. \[Kinvsandwich\].

We now define the network in Fig. \[Kinv\] by requiring that $K_1^{-1}$ act on a bra by inserting it into the first operand of that network, as illustrated in Fig. \[K1invaction\], and by requiring that $K_1^{-1}$ actually be the inverse of $K_1$. Figure \[K1invaction\] shows the action of $K_1^{-1}$ on an arbitrary bra $\bra{\phi}$. The use of the first operand of $K^{-1}$ for this purpose is a convention, 