The adjoint is defined for any [Linear operator](Linear%20operator.md), $A$ in a [complex vector space](Complex%20vector%20spaces.md), $\mathcal{V}$ equipped with an [inner product](Complex%20vector%20spaces.md#Complex%20inner%20product%20spaces) such that $\forall |u\rangle,|v\rangle\in\mathcal{V}$$\langle u, A v\rangle = \langle A^{\dagger}u, v\rangle.$ When acting on vector, $|u\rangle \in \mathcal{V}$, we obtain $|u\rangle^\dagger = \langle u |$. The vector $\langle u |$ is an element of the [complex dual space.](Complex%20vector%20spaces.md#Complex%20dual%20spaces) On a [real vector space](Real%20vector%20spaces.md) [the adjoint](Adjoint.md) is equivalent to the [transpose.](Transpose%20of%20a%20linear%20map.md) %%It needs to be addressed how the adjoint generalizes to infinite dimensional vector spaces and also unbounded operators. Address this by generalizing the below properties as well.%% # conjugate transpose The [adjoint](Adjoint.md) of an operator, when the operator may be written as a matrix is given by its conjugate [transpose](Transpose%20of%20a%20linear%20map.md#transpose%20of%20a%20matrix) (also called the _Hermitian transpose_). It is defined for an $m \times n$ matrix such that. $(A^\dagger)_{ij} = (A^*)_{ji}$ If all the matrix elements, $A_{ij}$ are real numbers then this is equivalent to the [transpose of a matrix](Transpose%20of%20a%20linear%20map.md#transpose%20of%20a%20matrix). ## Conjugate transpose Properties The conjugate transpose has the following properties. 1) $(AB)^{\dagger} = B^{\dagger}A^{\dagger}$ and more generally $(A...Z)^{\dagger} = Z^{\dagger}...A^{\dagger}$ ^01011f 2) The conjugate transpose is [anti-linear.](Anti-linear%20map.md) As such: $(A+B)^{\dagger}=(A^\dagger+B^\dagger)$ and $(cA)^\dagger=c^*A^\dagger$ where $c\in\mathbb{C}.$ ^6de25c #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/Matrices #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators