Do inner products show additivity in the first slot or second slot or both? In other words, which of the following is/are true?
$$\langle u,v+w\rangle = \langle u,v\rangle + \langle u,w\rangle$$ $$\langle u+v,w\rangle = \langle u,w\rangle + \langle v,w\rangle$$ $$ \langle u+v,w+x\rangle = \langle u,w\rangle + \langle u,x\rangle + \langle v,w\rangle + \langle v,x\rangle$$
$v,w,x,y \in$ Vector space $(V,F)$, $F = \mathbb{R}$ or $\mathbb{C}$
Note that inner products have the properties positive-definiteness, linearity in the first argument and conjugate symmetry, so it is indeed true that you always have $$\langle u+v,w\rangle = \langle u,w\rangle + \langle v,w\rangle$$ and $$\langle u,v+w\rangle = \overline{\langle v+w,u\rangle}=\overline{\langle v,u\rangle + \langle w,u\rangle}=\overline{\langle v,u\rangle} + \overline{\langle w,u\rangle}=\langle u,v\rangle + \langle u,w\rangle$$ and therefore also $$\langle u+v,w+x\rangle = \langle u,w+x\rangle + \langle v,w+x\rangle = \langle u,w\rangle + \langle u,x\rangle + \langle v,w\rangle + \langle v,x\rangle$$ for both $F=\mathbb{R}$ and $F=\mathbb{C}$.