Yeah, they are indeed just numbers. About the triangle inequality maybe it is something to do with the fact that if we take a vector which has component representation in an orthonormal basis of all ones:
$$\begin{bmatrix}
1 \\
1 \\
... \\
1
\end{bmatrix}$$
Then the inner product of this vector...
##| V_1 \rangle \in \mathbb{V}^{n_1}_1## and there is an orthonormal basis in ##\mathbb{V}^{n_1}_1##: ##|u_1\rangle, |u_2\rangle ... |u_{n_1}\rangle##
##| V_2 \rangle \in \mathbb{V}^{n_2}_2## and there is an orthonormal basis in ##\mathbb{V}^{n_2}_2##: ##|w_1\rangle, |w_2\rangle ...