KET AND BRA SPACE

The 2D vector space that was defined previously can also be called ket space. However, for every ket space, there also exists a bra space comprising of vectors that are complex conjugates of the same vectors called bra vectors or bras, and such a vector space is also called “dual space” or bra space.
This implies that for every ket (|↑〉_z) there exists a corresponding bra (〈​↑|_z) which is simply a complex conjugate of the ket.

.......(1)

........(2)

The spin-up and spin-down states are also called eigenkets of the spin operator ˆ S_z.