Unitary fusion paths form an orthonormal Hilbert basis

For $n=0$, $T_0=b$. If $b\cong ̸c$, simplicity gives $\mathrm{Hom}(c,b)=0$ and there are no paths. If $b\cong c$, the only empty-path vector is the chosen unitary isomorphism $u_{c,b}:c\to b$, and $u_{c,b}^{†}{u}_{c,b}={\mathrm{id}}_c$ shows it has norm one. Such a choice exists: for any isomorphism $v:c\to b$, [F3] gives $\mathrm{End}(c)=\mathbb{C}{\mathrm{id}}_c$, so $v^{†}v=\lambda {\mathrm{id}}_c$. Positivity in [F2] gives $\lambda \ge 0$, and $v^{†}v=0$ would imply $v=0$ by [F2]; hence $\lambda >0$. The rescaled map $u_{c,b}={\lambda }^{-1\mathrm{/}2}v$ satisfies $u_{c,b}^{†}{u}_{c,b}={\mathrm{id}}_c$. Since it is invertible, $u_{c,b}^{†}=u_{c,b}^{-1}$, so it is unitary. Thus the length-zero path family is orthonormal. For $n=1$, the path family is the given orthonormal split vertex basis $B_{b,a}^c$ itself, so the positive-length base case holds independently of the empty-path choice.

Fix $n\ge 1$ and assume that all length-$n$ path composites, for every simple endpoint, are orthonormal in their Hom spaces.

If $f_{\mu }:x\otimes a\to y$ are the chosen orthonormal vertex maps, their adjoints $i_{\mu }=f_{\mu }^{†}:y\to x\otimes a$ satisfy $i_{\mu }^{†}{i}_{\nu }=f_{\mu }{f}_{\nu }^{†}={\delta }_{\mu \nu }{\mathrm{id}}_y$ by [F3]. Distinct simple outputs are orthogonal, and the orthogonal semisimple decomposition has projections $i_{\mu }{i}_{\mu }^{†}$ summing to the identity.

Tensoring the orthonormal length-$n$ path inclusions by ${\mathrm{id}}_a$ preserves their adjoint products because $(f\otimes {\mathrm{id}}_a{)}^{†}=f^{†}\otimes {\mathrm{id}}_a$ by [F2]. Composing with the isometric inclusions of step 1.3 therefore gives orthonormal path inclusions at length $n+1$. Induction from step 1.1 proves orthonormality for every finite $n$.

By induction, every path composite is orthonormal. The basis lemma [F4] says the same finite family spans $H_{a,n;b,c}$; if there are no admissible paths, that space is zero and the empty family is its basis. Thus the path composites are an orthonormal basis for each $n$, completing induction from step 1.1.