Why is $\langle f+g, f+g \rangle = \langle f, f \rangle + \langle g, f \rangle + \langle f, g \rangle + \langle g,g\rangle$?

Question

Why is $\langle f+g, f+g \rangle = \langle f, f \rangle + \langle g, f \rangle + \langle f, g \rangle + \langle g,g\rangle$?

I know that from linearity:

$$\langle f+g, f+g \rangle = \langle f, f+g \rangle + \langle g, f+g \rangle$$

How to proceed?

Side note: It's very tedious writing langle/rangle again and again.

Answer 1

Using only the axioms of inner products (and that $\overline{a+b} = \overline{a}+\overline{b}$ ($*$)): \begin{align} \langle f+g,f+g\rangle &= \langle f,f+g\rangle+\langle g,f+g\rangle &\text{(Linearity in $1$st entry)}\\ &=\overline{\langle f+g,f\rangle} + \overline{\langle f+g,g\rangle}&\text{(Conjugate symmetry)}\\ &=\overline{\langle f,f\rangle+\langle g,f\rangle} + \overline{\langle f,g\rangle+\langle g,g\rangle}&\text{(Linearity in $1$st entry)}\\ &=\overline{\langle f,f\rangle} + \overline{\langle g,f\rangle} + \overline{\langle f,g\rangle}+\overline{\langle g,g\rangle}&\text{($*$)}\\ &= \langle f,f\rangle + \langle f,g\rangle + \langle g,f\rangle + \langle g,g\rangle.&\text{(Conjugate symmetry)} \end{align}

Answer 2

For symmetric inner product, you do not only have linearity but bilinearity. You can use linearity in the second argument to see

$$\langle f, f+g \rangle = \langle f, f \rangle + \langle f, g \rangle$$ $$\langle g, f+g \rangle = \langle g, f \rangle + \langle g, g \rangle$$