# Outer Products

\(|\cdot\rangle\langle \cdot|\). 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 \(\mathcal{V}\) (Specifically on span{ket})

### Specifically

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

### Note

\((|a\rangle\langle b)^\dagger = |b\rangle \langle a|\).