Let $C[-1,2]$ be the space of all continuous functions $f:[-1,2]\to \mathbb{C}$.
Which of the following define an inner product on $C[-1,2]$ and which do not?
$\langle f,g\rangle =\int_{-1}^{2}f(t)\overline{g(t)}dt+f(-\frac{1}{2})\overline{g(-\frac{1}{2})}$
$\langle f,g\rangle =3\int_{-1}^{2}f(t)\overline{g(t)}dt$
$\langle f,g\rangle =f(0)\overline{g(0)}+f(1)\overline{g(1)}$
It seems that all are inner products.
I am just not sure in the case of 1 if it is linear in the first argument
and in case 3, if it is an inner product can we conclude that $\langle f,g\rangle =c$ where $c\in \mathbb{R^{+}}$?