The condition that the scalar product of $\langle B| \;\text{and}\; |A\rangle$ is a linear function, is expressed by the equations:

Assume we have some number $\phi$, which is a linear function of $|A\rangle$, meaning for each ket, there is a corresponding $\phi$, then we say that the $\phi$ corresponding to $\small |A\rangle + |A^{\prime}\rangle$, is the sum of the numbers corresponding to $|A\rangle \;\text{and}\; |A^{\prime}\rangle$, and also, the number associated with $c|A\rangle$, is $c(\phi)$, for the $\phi$ associated with $|A\rangle$.