Editing Riesz representation theorem (section)

== Riesz representation theorem ==

Two vectors <math>x</math> and <math>y</math> are {{em|[[Orthogonal vectors|orthogonal]]}} if <math>\langle x, y \rangle = 0,</math> which happens if and only if <math>\|y\| \leq \|y + s x\|</math> for all scalars <math>s.</math>{{sfn|Rudin|1991|pp=306-312}} The [[orthogonal complement]] of a subset <math>X \subseteq H</math> is 
<math display=block>X^{\bot} := \{ \,y \in H : \langle y, x \rangle = 0 \text{ for all } x \in X\, \},</math>
which is always a [[closed set|closed]] vector subspace of <math>H.</math> 
The [[Hilbert projection theorem]] guarantees that for any [[NonEmpty|nonempty]] closed [[convex subset]] <math>C</math> of a [[Hilbert space]] there exists a unique vector <math>m \in C</math> such that <math>\|m\| = \inf_{c \in C} \|c\|;</math> that is, <math>m \in C</math> is the (unique) [[global minimum point]] of the function <math>C \to [0, \infty)</math> defined by <math>c \mapsto \|c\|.</math>

===Statement===

{{Math theorem
|name = {{visible anchor|Riesz representation theorem}}
|math_statement =
Let <math>H</math> be a [[Hilbert space]] whose [[inner product]] <math>\left\langle x, y \right\rangle</math> is linear in its {{em|first}} argument and [[Antilinear map|antilinear]] in its second argument and let <math>\langle y \mid x \rangle := \langle x, y \rangle</math> be the corresponding physics notation. For every continuous linear functional <math>\varphi \in H^*,</math> there exists a unique vector <math>f_{\varphi} \in H,</math> called the {{em|{{visible anchor|Riesz representation|Riesz representative}} of <math>\varphi,</math>}}<!-- the term "Riesz representation of" seems to be more commonly used than "Riesz representative of" --> such that<ref>{{harvnb|Roman|2008|loc=p. 351 Theorem 13.32}}</ref>
<math display=block>\varphi(x) = \left\langle x, f_{\varphi} \right\rangle = \left\langle f_\varphi \mid x \right\rangle \quad \text{ for all } x \in H.</math>

Importantly for {{em|complex}} Hilbert spaces, <math>f_{\varphi}</math> is always located in the {{em|antilinear}} coordinate of the inner product.<ref name="ImportanceOfLocationOfRieszRep" group="note" />

Furthermore, the length of the representation vector is equal to the norm of the functional:
<math display=block>\left\|f_\varphi\right\|_H = \|\varphi\|_{H^*},</math> and <math>f_{\varphi}</math> is the unique vector <math>f_{\varphi} \in \left(\ker \varphi\right)^{\bot}</math> with <math>\varphi\left(f_{\varphi}\right) = \|\varphi\|^2.</math> 
It is also the unique element of minimum norm in <math>C := \varphi^{-1}\left(\|\varphi\|^2\right)</math>; that is to say, <math>f_{\varphi}</math> is the unique element of <math>C</math> satisfying <math>\left\|f_{\varphi}\right\| = \inf_{c \in C} \|c\|.</math>
Moreover, any non-zero <math>q \in (\ker \varphi)^{\bot}</math> can be written as <math>q = \left(\|q\|^2 /\, \overline{\varphi(q)}\right)\  f_{\varphi}.</math>
}}

{{Math theorem
| name = Corollary
| math_statement = The {{em|canonical map from <math>H</math> into its dual}} <math>H^*</math>{{sfn|Trèves|2006|pp=112–123}} is the [[Injective map|injective]] [[Antilinear map|{{em|anti}}linear operator]] [[isometry]]<ref group=note name="AntilinearIsometryDef" />{{sfn|Trèves|2006|pp=112–123}} 
<math display=block>\begin{alignat}{4}
\Phi :\;&& H &&\;\to    \;& H^* \\[0.3ex]
        && y &&\;\mapsto\;& \langle \,\cdot\,, y \rangle = \langle y | \,\cdot\, \rangle \\
\end{alignat}</math>
The Riesz representation theorem states that this map is [[Surjective map|surjective]] (and thus [[Bijective map|bijective]]) when <math>H</math> is complete and that its inverse is the [[Bijective map|bijective]] [[Isometry|isometric]] antilinear isomorphism
<math display=block>\begin{alignat}{4}
\Phi^{-1} :\;&& H^*     &&\;\to    \;& H \\[0.3ex]
             && \varphi &&\;\mapsto\;& f_{\varphi} \\
\end{alignat}.</math>
Consequently, {{em|every}} continuous linear functional on the Hilbert space <math>H</math> can be written uniquely in the form <math>\langle y\, | \,\cdot\, \rangle</math>{{sfn|Trèves|2006|pp=112–123}} where <math>\|\langle y\,| \cdot \rangle\|_{H^*} = \|y\|_H</math> for every <math>y \in H.</math> 
The assignment <math>y \mapsto \langle y, \cdot \rangle = \langle \cdot\,| \,y \rangle</math> can also be viewed as a bijective {{em|linear}} isometry <math>H \to \overline{H}^*</math> into the [[Antilinear map|anti-dual space]] of <math>H,</math>{{sfn|Trèves|2006|pp=112–123}} which is the [[complex conjugate vector space]] of the [[continuous dual space]] <math>H^*.</math> 

The inner products on <math>H</math> and <math>H^*</math> are related by 
<math display=block>\left\langle \Phi h, \Phi k \right\rangle_{H^*} = \overline{\langle h, k \rangle}_H = \langle k, h \rangle_H \quad \text{ for all } h, k \in H</math>
and similarly, 
<math display=block>\left\langle \Phi^{-1} \varphi, \Phi^{-1} \psi \right\rangle_H = \overline{\langle \varphi, \psi \rangle}_{H^*} = \left\langle \psi, \varphi \right\rangle_{H^*} \quad \text{ for all } \varphi, \psi \in H^*.</math>

The set <math>C := \varphi^{-1}\left(\|\varphi\|^2\right)</math> satisfies <math>C = f_{\varphi} + \ker \varphi</math> and <math>C - f_{\varphi} = \ker \varphi</math> so when <math>f_{\varphi} \neq 0</math> then <math>C</math> can be interpreted as being the [[affine hyperplane]]<ref group=note name="VectorSpaceStructureOnAffineHyperplanesInducedByDualSpace" /> that is parallel to the vector subspace <math>\ker \varphi</math> and contains <math>f_{\varphi}.</math> 

For <math>y \in H,</math> the physics notation for the functional <math>\Phi(y) \in H^*</math> is the bra <math>\langle y |,</math> where explicitly this means that <math>\langle y | := \Phi(y),</math> which complements the ket notation <math>| y \rangle</math> defined by <math>| y \rangle := y.</math> 
In the mathematical treatment of [[quantum mechanics]], the theorem can be seen as a justification for the popular [[bra–ket notation]]. The theorem says that, every bra <math>\langle\psi\,|</math> has a corresponding ket <math>|\,\psi\rangle,</math> and the latter is unique.
}}

Historically, the theorem is often attributed simultaneously to [[Frigyes Riesz|Riesz]] and [[Maurice René Fréchet|Fréchet]] in 1907 (see references). 

{{collapse top|title=Proof{{sfn|Rudin|1991|pp=307−309}}|left=true}}
Let <math>\mathbb{F}</math> denote the underlying scalar field of <math>H.</math> 

{{em|Proof of norm formula:}} 

Fix <math>y \in H.</math>  
Define <math>\Lambda : H \to \mathbb{F}</math> by <math>\Lambda(z) := \langle \,y\, | \,z\, \rangle,</math> which is a linear functional on <math>H</math> since <math>z</math> is in the linear argument. 
By the [[Cauchy–Schwarz inequality]], 
<math display=block>|\Lambda(z)| = |\langle \,y\, | \,z\, \rangle| \leq \|y\| \|z\|</math>
which shows that <math>\Lambda</math> is bounded (equivalently, [[Continuous linear functional|continuous]]) and that <math>\|\Lambda\| \leq \|y\|.</math> 
It remains to show that <math>\|y\| \leq \|\Lambda\|.</math> 
By using <math>y</math> in place of <math>z,</math> it follows that 
<math display=block>\|y\|^2 = \langle \,y\, | \,y\, \rangle = \Lambda y = |\Lambda(y)| \leq \|\Lambda\| \|y\|</math>
(the equality <math>\Lambda y = |\Lambda(y)|</math> holds because <math>\Lambda y = \|y\|^2 \geq 0</math> is real and non-negative). 
Thus that <math>\|\Lambda\| = \|y\|.</math> 
<math>\blacksquare</math>

The proof above did not use the fact that <math>H</math> is [[Complete metric space|complete]], which shows that the formula for the norm <math>\|\langle \,y\, | \,\cdot\, \rangle\|_{H^*} = \|y\|_H</math> holds more generally for all [[inner product space]]s. 

{{hr|1}}

{{em|Proof that a Riesz representation of <math>\varphi</math> is unique:}} 

Suppose <math>f, g \in H</math> are such that <math>\varphi(z) = \langle \,f\, | \,z\, \rangle</math> and <math>\varphi(z) = \langle \,g\, | \,z\, \rangle</math> for all <math>z \in H.</math> 
Then 
<math display=block>\langle \,f - g\, | \,z\, \rangle = \langle \,f\, | \,z\, \rangle - \langle \,g\, | \,z\, \rangle = \varphi(z) - \varphi(z) = 0 \quad \text{ for all } z \in H</math>
which shows that <math>\Lambda := \langle \,f - g\, | \,\cdot\, \rangle</math> is the constant <math>0</math> linear functional. 
Consequently <math>0 = \|\langle \,f - g\, | \,\cdot\, \rangle\| = \|f - g\|,</math> which implies that <math>f - g = 0.</math> 
<math>\blacksquare</math>

{{hr|1}}

{{em|Proof that a vector <math>f_{\varphi}</math> representing <math>\varphi</math> exists:}} 

Let <math>K := \ker \varphi := \{ m \in H : \varphi(m) = 0 \}.</math> 
If <math>K = H</math> (or equivalently, if <math>\varphi = 0</math>) then taking <math>f_{\varphi} := 0</math> completes the proof so assume that <math>K \neq H</math> and <math>\varphi \neq 0.</math> 
The continuity of <math>\varphi</math> implies that <math>K</math> is a closed subspace of <math>H</math> (because <math>K = \varphi^{-1}(\{ 0 \})</math> and <math>\{ 0 \}</math> is a closed subset of <math>\mathbb{F}</math>). 
Let 
<math display=block>K^{\bot} := \{ v \in H ~:~ \langle \,v\, | \,k\, \rangle = 0 ~ \text{ for all } k \in K\}</math> 
denote the [[orthogonal complement]] of <math>K</math> in <math>H.</math>
Because <math>K</math> is closed and <math>H</math> is a Hilbert space,<ref group=note>Showing that there is a non-zero vector <math>v</math> in <math>K^{\bot}</math> relies on the continuity of <math>\phi</math> and the [[Cauchy completeness]] of <math>H.</math> This is the only place in the proof in which these properties are used.</ref> <math>H</math> can be written as the direct sum <math>H = K \oplus K^{\bot}</math><ref group=note>Technically, <math>H = K \oplus K^{\bot}</math> means that the addition map <math>K \times K^{\bot} \to H</math> defined by <math>(k, p) \mapsto k + p</math> is a surjective [[linear isomorphism]] and [[homeomorphism]]. See the article on [[complemented subspace]]s for more details.</ref> (a proof of this is given in the article on the [[Hilbert projection theorem]]). 
Because <math>K \neq H,</math> there exists some non-zero <math>p \in K^{\bot}.</math> 
For any <math>h \in H,</math>
<math display=block>\varphi[(\varphi h) p - (\varphi p) h] 
~=~ \varphi[(\varphi h) p] - \varphi[(\varphi p) h] 
~=~ (\varphi h) \varphi p - (\varphi p) \varphi h 
= 0,</math>
which shows that <math>(\varphi h) p - (\varphi p) h ~\in~ \ker \varphi = K,</math> where now <math>p \in K^{\bot}</math> implies 
<math display=block>0 = \langle \,p\, | \,(\varphi h) p - (\varphi p) h\, \rangle 
~=~ \langle \,p\, | \,(\varphi h) p \, \rangle - \langle \,p\, | \,(\varphi p) h\, \rangle 
~=~ (\varphi h) \langle \,p\, | \,p \, \rangle - (\varphi p) \langle \,p\, | \,h\, \rangle.</math> 
Solving for <math>\varphi h</math> shows that 
<math display=block>\varphi h 
= \frac{(\varphi p) \langle \,p\, | \,h\, \rangle}{\|p\|^2} 
= \left\langle \,\frac{\overline{\varphi p}}{\|p\|^2} p\, \Bigg| \,h\, \right\rangle \quad \text{ for every } h \in H,</math>
which proves that the vector <math>f_{\varphi} := \frac{\overline{\varphi p}}{\|p\|^2} p</math> satisfies 
<math>\varphi h = \langle \,f_{\varphi}\, | \,h\, \rangle \text{ for every } h \in H.</math> 

Applying the norm formula that was proved above with <math>y := f_{\varphi}</math> shows that <math>\|\varphi\|_{H^*} = \left\|\left\langle \,f_{\varphi}\, | \,\cdot\, \right\rangle\right\|_{H^*} = \left\|f_{\varphi}\right\|_H.</math> 
Also, the vector <math>u := \frac{p}{\|p\|}</math> has norm <math>\|u\| = 1</math> and satisfies <math>f_{\varphi} := \overline{\varphi(u)} u.</math> 
<math>\blacksquare</math>

{{hr|1}}

It can now be deduced that <math>K^{\bot}</math> is <math>1</math>-dimensional when <math>\varphi \neq 0.</math> 
Let <math>q \in K^{\bot}</math> be any non-zero vector. Replacing <math>p</math> with <math>q</math> in the proof above shows that the vector <math>g := \frac{\overline{\varphi q}}{\|q\|^2} q</math> satisfies <math>\varphi(h) = \langle \,g\, | \,h\, \rangle</math> for every <math>h \in H.</math> The uniqueness of the (non-zero) vector <math>f_{\varphi}</math> representing <math>\varphi</math> implies that <math>f_{\varphi} = g,</math> which in turn implies that <math>\overline{\varphi q} \neq 0</math> and <math>q = \frac{\|q\|^2}{\overline{\varphi q}} f_{\varphi}.</math> Thus every vector in <math>K^{\bot}</math> is a scalar multiple of <math>f_{\varphi}.</math> <math>\blacksquare</math>

The formulas for the inner products follow from the [[polarization identity]]. 

{{collapse bottom}}

=== Observations ===

If <math>\varphi \in H^*</math> then 
<math display=block>\varphi \left(f_{\varphi}\right) = \left\langle f_{\varphi}, f_{\varphi} \right\rangle = \left\|f_{\varphi}\right\|^2 = \|\varphi\|^2.</math> 
So in particular, <math>\varphi \left(f_{\varphi}\right) \geq 0</math> is always real and furthermore, <math>\varphi \left(f_{\varphi}\right) = 0</math> if and only if <math>f_{\varphi} = 0</math> if and only if <math>\varphi = 0.</math> 

'''Linear functionals as affine hyperplanes'''

A non-trivial continuous linear functional <math>\varphi</math> is often interpreted geometrically by identifying it with the affine hyperplane <math>A := \varphi^{-1}(1)</math> (the kernel <math>\ker\varphi = \varphi^{-1}(0)</math> is also often visualized alongside <math>A := \varphi^{-1}(1)</math> although knowing <math>A</math> is enough to reconstruct <math>\ker \varphi</math> because if <math>A = \varnothing</math> then <math>\ker \varphi = H</math> and otherwise <math>\ker \varphi = A - A</math>). In particular, the norm of <math>\varphi</math> should somehow be interpretable as the "norm of the hyperplane <math>A</math>". When <math>\varphi \neq 0</math> then the Riesz representation theorem provides such an interpretation of <math>\|\varphi\|</math> in terms of the affine hyperplane<ref group=note name="VectorSpaceStructureOnAffineHyperplanesInducedByDualSpace" /> 
<math>A := \varphi^{-1}(1)</math> as follows: using the notation from the theorem's statement, from <math>\|\varphi\|^2 \neq 0</math> it follows that <math>C := \varphi^{-1}\left(\|\varphi\|^2\right) = \|\varphi\|^2 \varphi^{-1}(1) = \|\varphi\|^2 A</math> and so <math>\|\varphi\| = \left\|f_{\varphi}\right\| = \inf_{c \in C} \|c\|</math> implies <math>\|\varphi\| = \inf_{a \in A} \|\varphi\|^2 \|a\|</math> and thus <math>\|\varphi\| = \frac{1}{\inf_{a \in A} \|a\|}.</math> 
This can also be seen by applying the [[Hilbert projection theorem]] to <math>A</math> and concluding that the global minimum point of the map <math>A \to [0, \infty)</math> defined by <math>a \mapsto \|a\|</math> is <math>\frac{f_{\varphi}}{\|\varphi\|^2} \in A.</math> 
The formulas 
<math display=block>\frac{1}{\inf_{a \in A} \|a\|} = \sup_{a \in A} \frac{1}{\|a\|}</math>
provide the promised interpretation of the linear functional's norm <math>\|\varphi\|</math> entirely in terms of its associated affine hyperplane <math>A = \varphi^{-1}(1)</math> (because with this formula, knowing only the {{em|set}} <math>A</math> is enough to describe the norm of its associated linear {{em|functional}}). Defining <math>\frac{1}{\infty} := 0,</math> the [[infimum]] formula 
<math display=block>\|\varphi\| = \frac{1}{\inf_{a \in \varphi^{-1}(1)} \|a\|}</math>
will also hold when <math>\varphi = 0.</math> 
When the supremum is taken in <math>\R</math> (as is typically assumed), then the supremum of the empty set is <math>\sup \varnothing = - \infty</math> but if the supremum is taken in the non-negative reals <math>[0, \infty)</math> (which is the [[Image of a function|image]]/range of the norm <math>\|\,\cdot\,\|</math> when <math>\dim H > 0</math>) then this supremum is instead <math>\sup \varnothing = 0,</math> in which case the supremum formula <math>\|\varphi\| = \sup_{a \in \varphi^{-1}(1)} \frac{1}{\|a\|}</math> will also hold when <math>\varphi = 0</math> (although the atypical equality <math>\sup \varnothing = 0</math> is usually unexpected and so risks causing confusion).

=== Constructions of the representing vector ===

Using the notation from the theorem above, several ways of constructing <math>f_{\varphi}</math> from <math>\varphi \in H^*</math> are now described. 
If <math>\varphi = 0</math> then <math>f_{\varphi} := 0</math>; in other words, 
<math display=block>f_0 = 0.</math>

This special case of <math>\varphi = 0</math> is henceforth assumed to be known, which is why some of the constructions given below start by assuming <math>\varphi \neq 0.</math> 

'''Orthogonal complement of kernel'''

If <math>\varphi \neq 0</math> then for any <math>0 \neq u \in (\ker\varphi)^{\bot},</math> 
<math display=block>f_{\varphi} := \frac{\overline{\varphi(u)} u}{\|u\|^2}.</math>

If <math>u \in (\ker\varphi)^{\bot}</math> is a [[unit vector]] (meaning <math>\|u\| = 1</math>) then 
<math display=block>f_{\varphi} := \overline{\varphi(u)} u</math>
(this is true even if <math>\varphi = 0</math> because in this case <math>f_{\varphi} = \overline{\varphi(u)} u = \overline{0} u = 0</math>). 
If <math>u</math> is a unit vector satisfying the above condition then the same is true of <math>-u,</math> which is also a unit vector in <math>(\ker\varphi)^{\bot}.</math> However, <math>\overline{\varphi(-u)} (-u) = \overline{\varphi(u)} u = f_\varphi</math> so both these vectors result in the same <math>f_{\varphi}.</math>

'''Orthogonal projection onto kernel'''

If <math>x \in H</math> is such that <math>\varphi(x) \neq 0</math> and if <math>x_K</math> is the [[orthogonal projection]] of <math>x</math> onto <math>\ker\varphi</math> then<ref group=proof name="FormulaOrthoProjectionKernel" />
<math display=block>f_{\varphi} = \frac{\|\varphi\|^2}{\varphi(x)} \left(x - x_K\right).</math>

'''Orthonormal basis'''

Given an [[orthonormal basis]] <math>\left\{e_i\right\}_{i \in I}</math> of <math>H</math> and a continuous linear functional <math>\varphi \in H^*,</math> the vector <math>f_{\varphi} \in H</math> can be constructed uniquely by 
<math display=block>f_\varphi = \sum_{i \in I} \overline{\varphi\left(e_i\right)} e_i</math> 
where all but at most countably many <math>\varphi\left(e_i\right)</math> will be equal to <math>0</math> and where the value of <math>f_{\varphi}</math> does not actually depend on choice of orthonormal basis (that is, using any other orthonormal basis for <math>H</math> will result in the same vector). 
If <math>y \in H</math> is written as <math>y = \sum_{i \in I} a_i e_i</math> then 
<math display=block>\varphi(y) = \sum_{i \in I} \varphi\left(e_i\right) a_i = \langle f_{\varphi} | y \rangle</math>
and
<math display=block>\left\|f_{\varphi}\right\|^2 = \varphi\left(f_{\varphi}\right) = \sum_{i \in I} \varphi\left(e_i\right) \overline{\varphi\left(e_i\right)} = \sum_{i \in I} \left|\varphi\left(e_i\right)\right|^2 = \|\varphi\|^2.</math>

If the orthonormal basis <math>\left\{e_i\right\}_{i \in I} = \left\{e_i\right\}_{i=1}^{\infty}</math> is a sequence then this becomes 
<math display=block>f_\varphi = \overline{\varphi\left(e_1\right)} e_1 + \overline{\varphi\left(e_2\right)} e_2 + \cdots </math> 
and if <math>y \in H</math> is written as <math>y = \sum_{i \in I} a_i e_i = a_1 e_1 + a_2 e_2 + \cdots</math> then 
<math display=block>\varphi(y) = \varphi\left(e_1\right) a_1 + \varphi\left(e_2\right) a_2 + \cdots = \langle f_{\varphi} | y \rangle.</math>

==== Example in finite dimensions using matrix transformations ====

Consider the special case of <math>H = \Complex^n</math> (where <math>n > 0</math> is an integer) with the standard inner product 
<math display=block>\langle z \mid w \rangle := \overline{\,\vec{z}\,\,}^{\operatorname{T}} \vec{w} \qquad \text{ for all } \; w, z \in H</math> 
where <math>w \text{ and } z</math> are represented as [[Column matrix|column matrices]] <math>\vec{w} := \begin{bmatrix}w_1 \\ \vdots \\ w_n\end{bmatrix}</math> and <math>\vec{z} := \begin{bmatrix}z_1 \\ \vdots \\ z_n\end{bmatrix}</math> with respect to the standard orthonormal basis <math>e_1, \ldots, e_n</math> on <math>H</math> (here, <math>e_i</math> is <math>1</math> at its <math>i</math><sup>th</sup> coordinate and <math>0</math> everywhere else; as usual, <math>H^*</math> will now be associated with the [[dual basis]]) and where <math>\overline{\,\vec{z}\,}^{\operatorname{T}} := \left[\overline{z_1}, \ldots, \overline{z_n}\right]</math> denotes the [[conjugate transpose]] of <math>\vec{z}.</math> 
Let <math>\varphi \in H^*</math> be any linear functional and let <math>\varphi_1, \ldots, \varphi_n \in \Complex</math> be the unique scalars such that 
<math display=block>\varphi\left(w_1, \ldots, w_n\right) = \varphi_1 w_1 + \cdots + \varphi_n w_n \qquad \text{ for all } \; w := \left(w_1, \ldots, w_n\right) \in H,</math> 
where it can be shown that <math>\varphi_i = \varphi\left(e_i\right)</math> for all <math>i = 1, \ldots, n.</math> 
Then the Riesz representation of <math>\varphi</math> is the vector 
<math display=block>f_{\varphi} ~:=~ \overline{\varphi_1} e_1 + \cdots + \overline{\varphi_n} e_n
~=~ \left(\overline{\varphi_1}, \ldots, \overline{\varphi_n}\right) 
\in H.</math>
To see why, identify every vector <math>w = \left(w_1, \ldots, w_n\right)</math> in <math>H</math> with the column matrix 
<math>\vec{w} := \begin{bmatrix}w_1 \\ \vdots \\ w_n\end{bmatrix}</math> 
so that <math>f_{\varphi}</math> is identified with <math>\vec{f_{\varphi}} := \begin{bmatrix}\overline{\varphi_1} \\ \vdots \\ \overline{\varphi_n}\end{bmatrix} =  \begin{bmatrix}\overline{\varphi\left(e_1\right)} \\ \vdots \\ \overline{\varphi\left(e_n\right)}\end{bmatrix}.</math> 
As usual, also identify the linear functional <math>\varphi</math> with its [[transformation matrix]], which is the [[row matrix]] <math>\vec{\varphi} := \left[\varphi_1, \ldots, \varphi_n\right]</math> so that <math>\vec{f_{\varphi}} := \overline{\,\vec{\varphi}\,\,}^{\operatorname{T}}</math> and the function <math>\varphi</math> is the assignment <math>\vec{w} \mapsto \vec{\varphi} \, \vec{w},</math> where the right hand side is [[matrix multiplication]]. Then for all <math>w = \left(w_1, \ldots, w_n\right) \in H,</math>
<math display=block>\varphi(w) 
= \varphi_1 w_1 + \cdots + \varphi_n w_n 
= \left[\varphi_1, \ldots, \varphi_n\right] 
\begin{bmatrix}w_1 \\ \vdots \\ w_n\end{bmatrix} 
= \overline{\begin{bmatrix}\overline{\varphi_1} \\ \vdots \\ \overline{\varphi_n}\end{bmatrix}}^{\operatorname{T}} 
\vec{w} 
= \overline{\,\vec{f_{\varphi}}\,\,}^{\operatorname{T}} \vec{w} 
= \left\langle \,\,f_{\varphi}\, \mid \,w\, \right\rangle,
</math>
which shows that <math>f_{\varphi}</math> satisfies the defining condition of the Riesz representation of <math>\varphi.</math> 
The bijective antilinear isometry <math>\Phi : H \to H^*</math> defined in the corollary to the Riesz representation theorem is the assignment that sends <math>z = \left(z_1, \ldots, z_n\right) \in H</math> to the linear functional <math>\Phi(z) \in H^*</math> on <math>H</math> defined by 
<math display=block>w = \left(w_1, \ldots, w_n\right) ~\mapsto~ \langle \,z\, \mid \,w\,\rangle = \overline{z_1} w_1 + \cdots + \overline{z_n} w_n,</math> 
where under the identification of vectors in <math>H</math> with column matrices and vector in <math>H^*</math> with row matrices, <math>\Phi</math> is just the assignment 
<math display=block>\vec{z} = \begin{bmatrix}z_1 \\ \vdots \\ z_n\end{bmatrix} ~\mapsto~ \overline{\,\vec{z}\,}^{\operatorname{T}} = \left[\overline{z_1}, \ldots, \overline{z_n}\right].</math> 
As described in the corollary, <math>\Phi</math>'s inverse <math>\Phi^{-1} : H^* \to H</math> is the antilinear isometry <math>\varphi \mapsto f_{\varphi},</math> which was just shown above to be: 
<math display=block>\varphi ~\mapsto~ f_{\varphi} ~:=~ \left(\overline{\varphi\left(e_1\right)}, \ldots, \overline{\varphi\left(e_n\right)}\right);</math> 
where in terms of matrices, <math>\Phi^{-1}</math> is the assignment 
<math display=block>\vec{\varphi} = \left[\varphi_1, \ldots, \varphi_n\right] ~\mapsto~ \overline{\,\vec{\varphi}\,\,}^{\operatorname{T}} = \begin{bmatrix}\overline{\varphi_1} \\ \vdots \\ \overline{\varphi_n}\end{bmatrix}.</math> 
Thus in terms of matrices, each of <math>\Phi : H \to H^*</math> and <math>\Phi^{-1} : H^* \to H</math> is just the operation of [[Conjugate transpose|conjugate transposition]] <math>\vec{v} \mapsto \overline{\,\vec{v}\,}^{\operatorname{T}}</math> (although between different spaces of matrices: if <math>H</math> is identified with the space of all column (respectively, row) matrices then <math>H^*</math> is identified with the space of all row (respectively, column matrices). 

This example used the standard inner product, which is the map <math>\langle z \mid w \rangle := \overline{\,\vec{z}\,\,}^{\operatorname{T}} \vec{w},</math> but if a different inner product is used, such as <math>\langle z \mid w \rangle_M := \overline{\,\vec{z}\,\,}^{\operatorname{T}} \, M \, \vec{w} \,</math> where <math>M</math> is any [[Hermitian matrix|Hermitian]] [[positive-definite matrix]], or if a different orthonormal basis is used then the transformation matrices, and thus also the above formulas, will be different.

=== Relationship with the associated real Hilbert space ===
{{See also|Complexification}}

Assume that <math>H</math> is a complex Hilbert space with inner product <math>\langle \,\cdot\mid\cdot\, \rangle.</math> 
When the Hilbert space <math>H</math> is reinterpreted as a real Hilbert space then it will be denoted by <math>H_{\R},</math> where the (real) inner-product on <math>H_{\R}</math> is the real part of <math>H</math>'s inner product; that is: 
<math display=block>\langle x, y \rangle_{\R} := \operatorname{re} \langle x, y \rangle.</math> 

The norm on <math>H_{\R}</math> induced by <math>\langle \,\cdot\,, \,\cdot\, \rangle_{\R}</math> is equal to the original norm on <math>H</math> and the continuous dual space of <math>H_{\R}</math> is the set of all {{em|real}}-valued bounded <math>\R</math>-linear functionals on <math>H_{\R}</math> (see the article about the [[polarization identity]] for additional details about this relationship). 
Let <math>\psi_{\R} := \operatorname{re} \psi</math> and <math>\psi_{i} := \operatorname{im} \psi</math> denote the real and imaginary parts of a linear functional <math>\psi,</math> so that <math>\psi = \operatorname{re} \psi + i \operatorname{im} \psi = \psi_{\R} + i \psi_{i}.</math> 
The formula [[Real and imaginary parts of a linear functional|expressing a linear functional]] in terms of its real part is
<math display=block>\psi(h) = \psi_{\R}(h) - i \psi_{\R} (i h) \quad \text{ for } h \in H,</math>
where <math>\psi_{i}(h) = - i \psi_{\R} (i h)</math> for all <math>h \in H.</math> 
It follows that <math>\ker\psi_{\R} = \psi^{-1}(i \R),</math> and that <math>\psi = 0</math> if and only if <math>\psi_{\R} = 0.</math> 
It can also be shown that <math>\|\psi\| = \left\|\psi_{\R}\right\| = \left\|\psi_i\right\|</math> where <math>\left\|\psi_{\R}\right\| := \sup_{\|h\| \leq 1} \left|\psi_{\R}(h)\right|</math> and <math>\left\|\psi_i\right\| := \sup_{\|h\| \leq 1} \left|\psi_i(h)\right|</math> are the usual [[operator norm]]s. 
In particular, a linear functional <math>\psi</math> is bounded if and only if its real part <math>\psi_{\R}</math> is bounded. 

'''Representing a functional and its real part'''

The Riesz representation of a continuous linear function <math>\varphi</math> on a complex Hilbert space is equal to the Riesz representation of its real part <math>\operatorname{re} \varphi</math> on its associated real Hilbert space. 

Explicitly, let <math>\varphi \in H^*</math> and as above, let <math>f_\varphi \in H</math> be the Riesz representation of <math>\varphi</math> obtained in <math>(H, \langle, \cdot, \cdot \rangle),</math> so it is the unique vector that satisfies <math>\varphi(x) = \left\langle f_{\varphi} \mid x \right\rangle</math> for all <math>x \in H.</math> 
The real part of <math>\varphi</math> is a continuous real linear functional on <math>H_{\R}</math> and so the Riesz representation theorem may be applied to <math>\varphi_{\R} := \operatorname{re} \varphi</math> and the associated real Hilbert space <math>\left(H_{\R}, \langle, \cdot, \cdot \rangle_{\R}\right)</math> to produce its Riesz representation, which will be denoted by <math>f_{\varphi_{\R}}.</math> 
That is, <math>f_{\varphi_{\R}}</math> is the unique vector in <math>H_{\R}</math> that satisfies <math>\varphi_{\R}(x) = \left\langle f_{\varphi_{\R}} \mid x \right\rangle_{\R}</math> for all <math>x \in H.</math> 
The conclusion is <math>f_{\varphi_{\R}} = f_{\varphi}.</math> 
This follows from the main theorem because <math>\ker\varphi_{\R} = \varphi^{-1}(i \R)</math> and if <math>x \in H</math> then 
<math display=block>\left\langle f_\varphi \mid x \right\rangle_{\R} = \operatorname{re} \left\langle f_\varphi \mid x \right\rangle = \operatorname{re} \varphi(x) = \varphi_{\R}(x)</math> 
and consequently, if <math>m \in \ker\varphi_{\R}</math> then <math>\left\langle f_{\varphi}\mid m \right\rangle_{\R} = 0,</math> which shows that <math>f_{\varphi} \in (\ker\varphi_{\R})^{\perp_{\R}}.</math> 
Moreover, <math>\varphi(f_\varphi) = \|\varphi\|^2</math> being a real number implies that <math>\varphi_{\R} (f_\varphi) = \operatorname{re} \varphi(f_\varphi) = \|\varphi\|^2.</math>
In other words, in the theorem and constructions above, if <math>H</math> is replaced with its real Hilbert space counterpart <math>H_{\R}</math> and if <math>\varphi</math> is replaced with <math>\operatorname{re} \varphi</math> then <math>f_{\varphi} = f_{\operatorname{re} \varphi}.</math> This means that vector <math>f_{\varphi}</math> obtained by using <math>\left(H_{\R}, \langle, \cdot, \cdot \rangle_{\R}\right)</math> and the real linear functional <math>\operatorname{re} \varphi</math> is the equal to the vector obtained by using the origin complex Hilbert space <math>\left(H, \left\langle, \cdot, \cdot \right\rangle\right)</math> and original complex linear functional <math>\varphi</math> (with identical norm values as well). 

Furthermore, if <math>\varphi \neq 0</math> then <math>f_{\varphi}</math> is perpendicular to <math>\ker\varphi_{\R}</math> with respect to <math>\langle \cdot, \cdot \rangle_{\R}</math> where the kernel of <math>\varphi</math> is be a ''proper'' subspace of the kernel of its real part <math>\varphi_{\R}.</math> Assume now that <math>\varphi \neq 0.</math> 
Then <math>f_{\varphi} \not\in \ker\varphi_{\R}</math> because <math>\varphi_{\R}\left(f_{\varphi}\right) = \varphi\left(f_{\varphi}\right) = \|\varphi\|^2 \neq 0</math> and <math>\ker\varphi</math> is a proper subset of <math>\ker\varphi_{\R}.</math> The vector subspace <math>\ker \varphi</math> has real codimension <math>1</math> in <math>\ker\varphi_{\R},</math> while <math>\ker\varphi_{\R}</math> has {{em|real}} codimension <math>1</math> in <math>H_{\R},</math> and <math>\left\langle f_{\varphi}, \ker\varphi_{\R} \right\rangle_{\R} = 0.</math> That is, <math>f_{\varphi}</math> is perpendicular to <math>\ker\varphi_{\R}</math> with respect to <math>\langle \cdot, \cdot \rangle_{\R}.</math>

=== Canonical injections into the dual and anti-dual ===

'''Induced linear map into anti-dual'''

The map defined by placing <math>y</math> into the {{em|linear}} coordinate of the inner product and letting the variable <math>h \in H</math> vary over the {{em|antilinear}} coordinate results in an [[Antilinear map|{{em|antilinear}} functional]]:
<math display=block>\langle \,\cdot \mid y\, \rangle = \langle \,y, \cdot\, \rangle : H \to \mathbb{F} \quad \text{ defined by } \quad h \mapsto \langle \,h \mid y\, \rangle = \langle \,y, h\, \rangle.</math>

This map is an element of <math>\overline{H}^*,</math> which is the continuous [[anti-dual space]] of <math>H.</math> 
The {{em|canonical map from <math>H</math> into its anti-dual}} <math>\overline{H}^*</math>{{sfn|Trèves|2006|pp=112–123}} is the [[Linear operator|{{em|linear}} operator]] 
<math display=block>\begin{alignat}{4}
\operatorname{In}_H^{\overline{H}^*} :\;&& H &&\;\to    \;& \overline{H}^* \\[0.3ex]
                                        && y &&\;\mapsto\;& \langle \,\cdot \mid y\, \rangle = \langle \,y, \cdot\, \rangle \\[0.3ex]
\end{alignat}</math>
which is also an [[Injective map|injective]] [[isometry]].{{sfn|Trèves|2006|pp=112–123}} 
The [[Fundamental theorem of Hilbert spaces]], which is related to Riesz representation theorem, states that this map is surjective (and thus [[Bijective map|bijective]]). Consequently, every antilinear functional on <math>H</math> can be written (uniquely) in this form.{{sfn|Trèves|2006|pp=112–123}} 

If <math>\operatorname{Cong} : H^* \to \overline{H}^*</math> is the canonical [[Antilinear map|{{em|anti}}linear]] [[Bijective map|bijective]] [[isometry]] <math>f \mapsto \overline{f}</math> that was defined above, then the following equality holds:
<math display=block>\operatorname{Cong} ~\circ~ \operatorname{In}_H^{H^*} ~=~ \operatorname{In}_H^{\overline{H}^*}.</math>