We denote a scalar product of two vectors $a, b$ in Hilbert space $H$ as $(a,b)$ or $\langle a, b\rangle$.

In Bra-Ket notation, we denote a vector $a$ in Hilbert space as $|a\rangle$. Also, we say that bras such as $\langle b|$ belong to the dual space $H^*$ . So Bras are linear transformations that map kets to a number.

Then it isn't always true that $\langle a \mid b\rangle=(a,b)$.

In quantum mechanics the above equality holds, is it that we define bra in such a way so that the above equality holds? Are there cases when the above equation is not true?