Outer Products

$| \cdot ⟩ ⟨ \cdot |$ . From a matrix representation, this is a matrix.

Definition

The outer product of a ket and a bra is formally a map that defines a linear operator on $V$ (Specifically on span{ket})

Specifically

Let $L (V)$ be the set of topologically linear operators on $V$ . Then the outer product is $(| \cdot ⟩, ⟨ \cdot |) : V \times V^{*} \to L (V)$ by $(| v ⟩, ⟨ w |) \to | v ⟩ ⟨ w |$ .

Note

$(| a ⟩ ⟨ b)^{†} = | b ⟩ ⟨ a |$ .