Editing Riesz representation theorem (section)

=== Definition of the adjoint ===

{{Main|Hermitian adjoint|Conjugate transpose}}

For every <math>z \in Z,</math> the scalar-valued map <math>\langle z\mid A (\cdot) \rangle_Z</math><ref group=note name="ExplicitDefOfInnerProductOfTranspose" /> on <math>H</math> defined by 
<math display=block>h \mapsto \langle z\mid A h \rangle_Z = \langle A h, z \rangle_Z</math> 

is a continuous linear functional on <math>H</math> and so by the Riesz representation theorem, there exists a unique vector in <math>H,</math> denoted by <math>A^* z,</math> such that <math>\langle z \mid A (\cdot) \rangle_Z = \left\langle A^* z \mid \cdot\, \right\rangle_H,</math> or equivalently, such that 
<math display=block>\langle z \mid A h \rangle_Z = \left\langle A^* z \mid h \right\rangle_H \quad \text{ for all } h \in H.</math> 

The assignment <math>z \mapsto A^* z</math> thus induces a function <math>A^* : Z \to H</math> called the {{em|adjoint}} of <math>A : H \to Z</math> whose defining condition is 
<math display=block>\langle z \mid A h \rangle_Z = \left\langle A^* z\mid h \right\rangle_H \quad \text{ for all } h \in H  \text{ and all } z \in Z.</math> 
The adjoint <math>A^* : Z \to H</math> is necessarily a [[Continuous linear operator|continuous]] (equivalently, a [[Bounded linear operator|bounded]]) [[linear operator]]. 

If <math>H</math> is finite dimensional with the standard inner product and if <math>M</math> is the [[transformation matrix]] of <math>A</math> with respect to the standard orthonormal basis then <math>M</math>'s [[conjugate transpose]] <math>\overline{M^{\operatorname{T}}}</math> is the transformation matrix of the adjoint <math>A^*.</math>