In Dirac notation, why do we have $\langle cf|g\rangle = c^*\langle f\vert g\rangle$?

Question

A Hilbert space's inner product has the property $$\langle cf,g\rangle=c\langle f,g\rangle$$ where $f$ and $g$ are the vectors in the Hilbert Space and $c$ is a complex number. In Dirac Notation: $$\langle cf|g\rangle = c^*\langle f\vert g\rangle$$ I am confused about why the Dirac notation takes complex conjugate on $c$. I know $\langle f \vert$ in the Dirac notation is in the dual Hilbert space. Is this the reason? Do I need to define a Hilbert Space as
$\langle cf,g\rangle=c\langle f,g\rangle$? I feel it is equivalent if I define as $\langle cf,g\rangle = c^*\langle f,g\rangle$? Am I right? Thanks a lot!

Answer 1

Inner products on Hilbert spaces are linear in their first argument and conjugate linear in their second argument. So Hilbert spaces also have the property $$\langle f, \, cg\rangle = c^\ast \langle f, \, g \rangle.$$

As you said, a bra represents the dual space to a ket. So to move to an inner product in Dirac notation simply note that for $c_1, \, c_2 \in \mathbb{C}$ and $f, \, g \in \mathcal{H}$ $$\langle c_1 f , \, c_2 g \rangle = \langle c_2 g \vert c_1 f \rangle = c_1 \, c_2^\ast \langle f, \, g \rangle.$$