Editing Riesz representation theorem (section)

=== Adjoints are transposes ===

{{Main|Transpose of a linear map}}
{{See also|Transpose}}

It is also possible to define the {{em|[[Transpose of a linear map|transpose]]}} or {{em|[[algebraic adjoint]]}} of <math>A : H \to Z,</math> which is the map <math>{}^{t}A : Z^* \to H^*</math> defined by sending a continuous linear functionals <math>\psi \in Z^*</math> to 
<math display=block>{}^{t}A(\psi) := \psi \circ A,</math> 
where the [[Function composition|composition]] <math>\psi \circ A</math> is always a continuous linear functional on <math>H</math> and it satisfies <math>\|A\| = \left\|{}^t A\right\|</math> (this is true more generally, when <math>H</math> and <math>Z</math> are merely [[normed space]]s).{{sfn|Rudin|1991|pp=92-115}} 
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which is equivalent to:
<math display=block>\psi(A h) = \left({}^t A(\psi)\right) h \quad \text{ for all } h \in H  \text{ and all } \psi \in Z^*.</math>
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So for example, if <math>z \in Z</math> then <math>{}^{t}A</math> sends the continuous linear functional <math>\langle z \mid \cdot \rangle_Z \in Z^*</math> (defined on <math>Z</math> by <math>g \mapsto \langle z \mid g \rangle_Z</math>) to the continuous linear functional <math>\langle z \mid A(\cdot) \rangle_Z \in H^*</math> (defined on <math>H</math> by <math>h \mapsto \langle z \mid A(h) \rangle_Z</math>);<ref group=note name="ExplicitDefOfInnerProductOfTranspose" /> 
using bra-ket notation, this can be written as <math>{}^{t}A \langle z \mid ~=~ \langle z \mid A</math> where the juxtaposition of <math>\langle z \mid</math> with <math>A</math> on the right hand side denotes function composition: <math>H \xrightarrow{A} Z \xrightarrow{\langle z \mid} \Complex.</math> 

The adjoint <math>A^* : Z \to H</math> is actually just to the transpose <math>{}^{t}A : Z^* \to H^*</math>{{sfn|Rudin|1991|pp=306-312}} when the Riesz representation theorem is used to identify <math>Z</math> with <math>Z^*</math> and <math>H</math> with <math>H^*.</math> 

Explicitly, the relationship between the adjoint and transpose is:

{{NumBlk|:|<math>{}^{t}A ~\circ~ \Phi_Z ~=~ \Phi_H ~\circ~ A^*</math>|{{EquationRef|Adjoint-transpose}}|LnSty=1px dashed black}}

which can be rewritten as:
<math display=block>A^* ~=~ \Phi_H^{-1} ~\circ~ {}^{t}A ~\circ~ \Phi_Z \quad \text{ and } \quad {}^{t}A ~=~ \Phi_H ~\circ~ A^* ~\circ~ \Phi_Z^{-1}.</math>

{{Math proof|title=Proof|drop=hidden|proof=
To show that <math>{}^{t}A ~\circ~ \Phi_Z ~=~ \Phi_H ~\circ~ A^*,</math> fix <math>z \in Z.</math> 
The definition of <math>{}^{t}A</math> implies <math display=block>\left({}^{t}A \circ \Phi_Z\right) z = {}^{t}A \left(\Phi_Z z\right) = \left(\Phi_Z z\right) \circ A</math> so it remains to show that <math>\left(\Phi_Z z\right) \circ A = \Phi_H\left(A^* z\right).</math> If <math>h \in H</math> then <math display=block>\left(\left(\Phi_Z z\right) \circ A\right) h = \left(\Phi_Z z\right)(A h) = \langle z\mid A h \rangle_Z = \langle A^* z\mid h \rangle_H = \left(\Phi_H(A^* z)\right) h,</math> as desired. <math>\blacksquare</math>
}}

Alternatively, the value of the left and right hand sides of ({{EquationNote|Adjoint-transpose}}) at any given <math>z \in Z</math> can be rewritten in terms of the inner products as: 
<math display=block>\left({}^{t}A ~\circ~ \Phi_Z\right) z = \langle z \mid A (\cdot) \rangle_Z \quad \text{ and } \quad\left(\Phi_H ~\circ~ A^*\right) z = \langle A^* z\mid\cdot\, \rangle_H</math>
so that <math>{}^{t}A ~\circ~ \Phi_Z ~=~ \Phi_H ~\circ~ A^*</math> holds if and only if <math>\langle z \mid A (\cdot) \rangle_Z = \langle A^* z\mid\cdot\, \rangle_H</math> holds; but the equality on the right holds by definition of <math>A^* z.</math> 
The defining condition of <math>A^* z</math> can also be written
<math display=block>\langle z \mid A ~=~ \langle A^*z \mid</math> 
if bra-ket notation is used.