The inner product is conjugate symmetric; that is, the inner product of a pair of elements is equal to the complex conjugate of the inner product of the swapped elements: ⟨ y , x ⟩ = ⟨ x , y ⟩ ¯ . {\displaystyle \langle y,x\rangle ={\overline {\langle x,y\rangle }}\,.} Importantly, this implies that ⟨ x , x ⟩ {\displaystyle \langle x,x\rangle } is a real number. The inner product is linear in its first[nb 1] argument.