View source for Riesz representation theorem

{{Short description|Theorem about the dual of a Hilbert space}}
{{about|a theorem concerning the dual of a Hilbert space|the theorems relating linear functionals to measures|Riesz–Markov–Kakutani representation theorem|other theorems|Riesz theorem (disambiguation){{!}}Riesz theorem}}

The '''Riesz representation theorem''', sometimes called the '''Riesz–Fréchet representation theorem''' after [[Frigyes Riesz]] and [[Maurice René Fréchet]], establishes an important connection between a [[Hilbert space]] and its [[continuous dual space]]. If the underlying [[Field (mathematics)|field]] is the [[real number]]s, the two are [[isometry|isometrically]] [[isomorphism|isomorphic]]; if the underlying field is the [[complex number]]s, the two are isometrically [[anti-isomorphic]]. The (anti-) [[isomorphism]] is a particular [[natural isomorphism]].

== Preliminaries and notation ==

Let <math>H</math> be a [[Hilbert space]] over a field <math>\mathbb{F},</math> where <math>\mathbb{F}</math> is either the real numbers <math>\R</math> or the complex numbers <math>\Complex.</math> If <math>\mathbb{F} = \Complex</math> (resp. if <math>\mathbb{F} = \R</math>) then <math>H</math> is called a {{em|complex Hilbert space}} (resp. a {{em|real Hilbert space}}). Every real Hilbert space can be extended to be a [[dense subset]] of a unique (up to [[Bijective map|bijective]] [[isometry]]) complex Hilbert space, called its [[complexification]], which is why Hilbert spaces are often automatically assumed to be complex. Real and complex Hilbert spaces have in common many, but by no means all, properties and results/theorems. 

This article is intended for both [[Mathematician|mathematicians]] and [[Physicist|physicists]] and will describe the theorem for both. 
In both mathematics and physics, if a Hilbert space is assumed to be real (that is, if <math>\mathbb{F} = \R</math>) then this will usually be made clear. Often in mathematics, and especially in physics, unless indicated otherwise, "Hilbert space" is usually automatically assumed to mean "complex Hilbert space." Depending on the author, in mathematics, "Hilbert space" usually means either (1) a complex Hilbert space, or (2) a real {{em|or}} complex Hilbert space. 

=== Linear and antilinear maps ===

By definition, an [[Antilinear map|{{em|antilinear map}}]] (also called a {{em|conjugate-linear map}}) <math>f : H \to Y</math> is a map between [[vector space]]s that is {{em|[[Additive map|additive]]}}:
<math display=block>f(x + y) = f(x) + f(y) \quad \text{ for all } x, y \in H,</math>
and {{em|antilinear}} (also called {{em|conjugate-linear}} or {{em|conjugate-homogeneous}}):
<math display="block">f(c x) = \overline{c} f(x) \quad \text{ for all } x \in H \text{ and all scalar } c \in \mathbb{F},</math>
where <math>\overline{c}</math> is the conjugate of the complex number <math>c = a + b i</math>, given by <math>\overline{c} = a - b i</math>.

In contrast, a map <math>f : H \to Y</math> is [[Linear map|linear]] if it is additive and [[Homogeneous function|{{em|homogeneous}}]]: 
<math display=block>f(c x) = c f(x) \quad \text{ for all } x \in H \quad \text{ and all scalars } c \in \mathbb{F}.</math>

Every constant <math>0</math> map is always both linear and antilinear. If <math>\mathbb{F} = \R</math> then the definitions of linear maps and antilinear maps are completely identical. A linear map from a Hilbert space into a [[Banach space]] (or more generally, from any Banach space into any [[topological vector space]]) is [[Continuous linear operator|continuous]] if and only if it is [[Bounded linear operator|bounded]]; the same is true of antilinear maps. The [[Inverse function|inverse]] of any antilinear (resp. linear) bijection is again an antilinear (resp. linear) bijection. The composition of two {{em|anti}}linear maps is a {{em|linear}} map. 

'''Continuous dual and anti-dual spaces'''

A {{em|functional}} on <math>H</math> is a function <math>H \to \mathbb{F}</math> whose [[codomain]] is the underlying scalar field <math>\mathbb{F}.</math> 
Denote by <math>H^*</math> (resp. by <math>\overline{H}^*)</math> the set of all continuous linear (resp. continuous antilinear) functionals on <math>H,</math> which is called the {{em|[[Continuous dual space|(continuous) dual space]]}} (resp. the {{em|[[Anti-dual space|(continuous) anti-dual space]]}}) of <math>H.</math>{{sfn|Trèves|2006|pp=112–123}} 
If <math>\mathbb{F} = \R</math> then linear functionals on <math>H</math> are the same as antilinear functionals and consequently, the same is true for such continuous maps: that is, <math>H^* = \overline{H}^*.</math> 

'''One-to-one correspondence between linear and antilinear functionals'''

Given any functional <math>f ~:~ H \to \mathbb{F},</math> the {{em|conjugate of <math>f</math>}} is the functional 
<math display=block>\begin{alignat}{4}
\overline{f} : \,& H && \to    \,&& \mathbb{F} \\
                 & h && \mapsto\,&& \overline{f(h)}. \\
\end{alignat}</math>

This assignment is most useful when <math>\mathbb{F} = \Complex</math> because if <math>\mathbb{F} = \R</math> then <math>f = \overline{f}</math> and the assignment <math>f \mapsto \overline{f}</math> reduces down to the [[identity map]]. 

The assignment <math>f \mapsto \overline{f}</math> defines an antilinear [[Bijective map|bijective]] correspondence from the set of 
:all functionals (resp. all linear functionals, all continuous linear functionals <math>H^*</math>) on <math>H,</math> 
onto the set of 
:all functionals (resp. all {{em|anti}}linear functionals, all continuous {{em|anti}}linear functionals <math>\overline{H}^*</math>) on <math>H.</math>

=== Mathematics vs. physics notations and definitions of inner product ===

The [[Hilbert space]] <math>H</math> has an associated [[inner product]] <math>H \times H \to \mathbb{F}</math> valued in <math>H</math>'s underlying scalar field <math>\mathbb{F}</math> that is linear in one coordinate and antilinear in the other (as specified below).
If <math>H</math> is a complex Hilbert space (<math>\mathbb{F} = \Complex</math>), then there is a crucial difference between the notations prevailing in mathematics versus physics, regarding which of the two variables is linear.
However, for real Hilbert spaces (<math>\mathbb{F} = \R</math>), the inner product is a [[Symmetric map|symmetric]] map that is linear in each coordinate ([[bilinear map|bilinear]]), so there can be no such confusion. 

In [[mathematics]], the inner product on a Hilbert space <math>H</math> is often denoted by <math>\left\langle \cdot\,, \cdot \right\rangle</math> or <math>\left\langle \cdot\,, \cdot \right\rangle_H</math> while in [[physics]], the [[bra–ket notation]] <math>\left\langle \cdot \mid \cdot \right\rangle</math> or <math>\left\langle \cdot \mid \cdot \right\rangle_H</math> is typically used. In this article, these two notations will be related by the equality:
<math display="block">\left\langle x, y \right\rangle := \left\langle y \mid x \right\rangle \quad \text{ for all } x, y \in H.</math>These have the following properties:<ol>
<li>The map <math>\left\langle \cdot\,, \cdot \right\rangle</math> is ''linear in its first coordinate''; equivalently, the map <math>\left\langle \cdot \mid \cdot \right\rangle</math> is ''linear in its second coordinate''. That is, for fixed <math>y \in H,</math> the map 
<math>\left\langle \,y\mid \cdot\, \right\rangle = \left\langle \,\cdot\,, y\, \right\rangle : H \to \mathbb{F}</math> with <math display="inline">h \mapsto \left\langle \,y\mid h\, \right\rangle = \left\langle \,h, y\, \right\rangle </math>
is a linear functional on <math>H.</math> This linear functional is continuous, so <math>\left\langle \,y\mid\cdot\, \right\rangle = \left\langle \,\cdot, y\, \right\rangle \in H^*.</math>
</li>
<li>The map <math>\left\langle \cdot\,, \cdot \right\rangle</math> is ''[[Antilinear map|antilinear]] in its {{em|second}} coordinate''; equivalently, the map <math>\left\langle \cdot \mid \cdot \right\rangle</math> is ''antilinear in its {{em|first}} coordinate''. That is, for fixed <math>y \in H,</math> the map
<math>\left\langle \,\cdot\mid y\, \right\rangle = \left\langle \,y, \cdot\, \right\rangle : H \to \mathbb{F}</math> with <math display="inline">h \mapsto \left\langle \,h\mid y\, \right\rangle = \left\langle \,y, h\, \right\rangle </math>
is an antilinear functional on <math>H.</math> This antilinear functional is continuous, so <math>\left\langle \,\cdot\mid y\, \right\rangle = \left\langle \,y, \cdot\, \right\rangle \in \overline{H}^*.</math>
</li>
</ol>

In computations, one must consistently use either the mathematics notation <math>\left\langle \cdot\,, \cdot \right\rangle</math>, which is (linear, antilinear); or the physics notation <math>\left\langle \cdot \mid \cdot \right\rangle</math>, which is (antilinear | linear).

=== Canonical norm and inner product on the dual space and anti-dual space ===

If <math>x = y</math> then <math>\langle \,x\mid x\, \rangle = \langle \,x, x\, \rangle</math> is a non-negative real number and the map
<math display=block>\|x\| := \sqrt{\langle x, x \rangle} = \sqrt{\langle x \mid x \rangle}</math> 

defines a [[Norm (mathematics)|canonical norm]] on <math>H</math> that makes <math>H</math> into a [[normed space]].{{sfn|Trèves|2006|pp=112–123}} 
As with all normed spaces, the (continuous) dual space <math>H^*</math> carries a canonical norm, called the {{em|[[dual norm]]}}, that is defined by{{sfn|Trèves|2006|pp=112–123}}
<math display=block>\|f\|_{H^*} ~:=~ \sup_{\|x\| \leq 1, x \in H} |f(x)| \quad \text{ for every } f \in H^*.</math> 

The canonical norm on the (continuous) [[anti-dual space]] <math>\overline{H}^*,</math> denoted by <math>\|f\|_{\overline{H}^*},</math> is defined by using this same equation:{{sfn|Trèves|2006|pp=112–123}} 
<math display=block>\|f\|_{\overline{H}^*} ~:=~ \sup_{\|x\| \leq 1, x \in H} |f(x)| \quad \text{ for every } f \in \overline{H}^*.</math> 

This canonical norm on <math>H^*</math> satisfies the [[parallelogram law]], which means that the [[polarization identity]] can be used to define a {{em|canonical inner product on <math>H^*,</math>}} which this article will denote by the notations 
<math display=block>\left\langle f, g \right\rangle_{H^*} := \left\langle g \mid f \right\rangle_{H^*},</math> 
where this inner product turns <math>H^*</math> into a Hilbert space. There are now two ways of defining a norm on <math>H^*:</math> the norm induced by this inner product (that is, the norm defined by <math>f \mapsto \sqrt{\left\langle f, f \right\rangle_{H^*}}</math>) and the usual [[dual norm]] (defined as the supremum over the closed unit ball). These norms are the same; explicitly, this means that the following holds for every <math>f \in H^*:</math>
<math display=block>\sup_{\|x\| \leq 1, x \in H} |f(x)| = \|f\|_{H^*} ~=~ \sqrt{\langle f, f \rangle_{H^*}} ~=~ \sqrt{\langle f \mid f \rangle_{H^*}}.</math> 

As will be described later, the Riesz representation theorem can be used to give an equivalent definition of the canonical norm and the canonical inner product on <math>H^*.</math> 

The same equations that were used above can also be used to define a norm and inner product on <math>H</math>'s [[anti-dual space]] <math>\overline{H}^*.</math>{{sfn|Trèves|2006|pp=112–123}} 

'''Canonical isometry between the dual and antidual'''

The [[complex conjugate]] <math>\overline{f}</math> of a functional <math>f,</math> which was defined above, satisfies 
<math display=block>\|f\|_{H^*} ~=~ \left\|\overline{f}\right\|_{\overline{H}^*} \quad \text{ and } \quad \left\|\overline{g}\right\|_{H^*} ~=~ \|g\|_{\overline{H}^*}</math>
for every <math>f \in H^*</math> and every <math>g \in \overline{H}^*.</math> 
This says exactly that the canonical antilinear [[Bijective map|bijection]] defined by
<math display=block>\begin{alignat}{4}
\operatorname{Cong} :\;&& H^* &&\;\to    \;& \overline{H}^* \\[0.3ex]
                       && f   &&\;\mapsto\;& \overline{f} \\
\end{alignat}</math>
as well as its inverse <math>\operatorname{Cong}^{-1} ~:~ \overline{H}^* \to H^*</math> are antilinear [[Isometry|isometries]] and consequently also [[homeomorphism]]s. 
The inner products on the dual space <math>H^*</math> and the anti-dual space <math>\overline{H}^*,</math> denoted respectively by <math>\langle \,\cdot\,, \,\cdot\, \rangle_{H^*}</math> and <math>\langle \,\cdot\,, \,\cdot\, \rangle_{\overline{H}^*},</math> are related by
<math display=block>\langle \,\overline{f}\, | \,\overline{g}\, \rangle_{\overline{H}^*} = \overline{\langle \,f\, | \,g\, \rangle_{H^*}} = \langle \,g\, | \,f\, \rangle_{H^*} \qquad \text{ for all } f, g \in H^*</math>
and
<math display=block>\langle \,\overline{f}\, | \,\overline{g}\, \rangle_{H^*} = \overline{\langle \,f\, | \,g\, \rangle_{\overline{H}^*}} = \langle \,g\, | \,f\, \rangle_{\overline{H}^*} \qquad \text{ for all } f, g \in \overline{H}^*.</math>

If <math>\mathbb{F} = \R</math> then <math>H^* = \overline{H}^*</math> and this canonical map <math>\operatorname{Cong} : H^* \to \overline{H}^*</math> reduces down to the identity map.

== 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>

== Extending the bra–ket notation to bras and kets ==

{{Main|Bra–ket notation}}

Let <math>\left(H, \langle\cdot, \cdot \rangle_H\right)</math> be a Hilbert space and as before, let <math>\langle y\, | \,x \rangle_H := \langle x, y \rangle_H.</math> 
Let 
<math display=block>\begin{alignat}{4}
\Phi :\;&& H &&\;\to    \;& H^* \\[0.3ex]
        && g &&\;\mapsto\;& \left\langle \,g\mid \cdot\, \right\rangle_H = \left\langle \,\cdot, g\, \right\rangle_H \\
\end{alignat}</math>
which is a bijective antilinear isometry that satisfies
<math display=block>(\Phi h) g = \langle h\mid g \rangle_H = \langle g, h \rangle_H \quad \text{ for all } g, h \in H.</math>

'''Bras'''

Given a vector <math>h \in H,</math> let <math>\langle h\, |</math> denote the continuous linear functional <math>\Phi h</math>; that is, 
<math display=block>\langle h\, | ~:=~ \Phi h</math> 
so that this functional <math>\langle h\, |</math> is defined by <math>g \mapsto \left\langle \,h\mid g\, \right\rangle_H.</math> This map was denoted by <math>\left\langle h \mid \cdot\, \right\rangle</math> earlier in this article. 

The assignment <math>h \mapsto \langle h |</math> is just the isometric antilinear isomorphism <math>\Phi ~:~ H \to H^*,</math> which is why <math>~\langle c g + h\, | ~=~ \overline{c} \langle g\mid ~+~ \langle h\, |~</math> holds for all <math>g, h \in H</math> and all scalars <math>c.</math> 
The result of plugging some given <math>g \in H</math> into the functional <math>\langle h\, |</math> is the scalar <math>\langle h\, | \,g \rangle_H = \langle g, h \rangle_H,</math> which may be denoted by <math>\langle h \mid g \rangle.</math><ref group=note>The usual notation for plugging an element <math>g</math> into a linear map <math>F</math> is <math>F(g)</math> and sometimes <math>Fg.</math> Replacing <math>F</math> with <math>\langle h\mid :=~ \Phi h</math> produces <math>\langle h\mid(g)</math> or <math>\langle h \mid g,</math> which is unsightly (despite being consistent with the usual notation used with functions). Consequently, the symbol <math>\,\rangle\,</math> is appended to the end, so that the notation <math>\langle h\mid g \rangle</math> is used instead to denote this value <math>(\Phi h) g.</math></ref> 

'''Bra of a linear functional'''

Given a continuous linear functional <math>\psi \in H^*,</math> let <math>\langle \psi\mid</math> denote the vector <math>\Phi^{-1} \psi \in H</math>; that is, 
<math display=block>\langle \psi\mid ~:=~ \Phi^{-1} \psi.</math> 

The assignment <math>\psi \mapsto \langle \psi\mid</math> is just the isometric antilinear isomorphism <math>\Phi^{-1}  ~:~ H^* \to H,</math> which is why <math>~\langle c \psi  + \phi\mid ~=~ \overline{c} \langle \psi\mid ~+~ \langle \phi\mid~</math> holds for all <math>\phi, \psi \in H^*</math> and all scalars <math>c.</math> 

The defining condition of the vector <math>\langle \psi | \in H</math> is the technically correct but unsightly equality
<math display=block>\left\langle \, \langle \psi\mid \, \mid g \right\rangle_H ~=~ \psi g \quad \text{ for all } g \in H,</math>
which is why the notation <math>\left\langle \psi \mid g \right\rangle</math> is used in place of <math>\left\langle \, \langle \psi\mid \, \mid g \right\rangle_H = \left\langle g, \, \langle \psi\mid \right\rangle_H.</math> With this notation, the defining condition becomes
<math display=block>\left\langle \psi\mid g \right\rangle ~=~ \psi g \quad \text{ for all } g \in H.</math>

'''Kets'''

For any given vector <math>g \in H,</math> the notation <math>| \,g \rangle</math> is used to denote <math>g</math>; that is, 
<math display=block>\mid g \rangle : = g.</math> 

The assignment <math>g \mapsto | \,g \rangle</math> is just the identity map <math>\operatorname{Id}_H : H \to H,</math> which is why <math>~\mid c g + h \rangle ~=~ c \mid g \rangle ~+~ \mid h \rangle~</math> holds for all <math>g, h \in H</math> and all scalars <math>c.</math> 

The notation <math>\langle h\mid g \rangle</math> and <math>\langle \psi\mid g \rangle</math> is used in place of <math>\left\langle h\mid \, \mid g \rangle \, \right\rangle_H ~=~ \left\langle \mid g \rangle, h \right\rangle_H</math> and <math>\left\langle \psi\mid \, \mid g \rangle \, \right\rangle_H ~=~ \left\langle g, \, \langle \psi\mid \right\rangle_H,</math> respectively. As expected, <math>~\langle \psi\mid g \rangle = \psi g~</math> and <math>~\langle h\mid g \rangle~</math> really is just the scalar <math>~\langle h\mid g \rangle_H ~=~ \langle g, h \rangle_H.</math>

== Adjoints and transposes ==

Let <math>A : H \to Z</math> be a [[continuous linear operator]] between [[Hilbert space|Hilbert spaces]] <math>\left(H, \langle \cdot, \cdot \rangle_H\right)</math> and <math>\left(Z, \langle \cdot, \cdot \rangle_Z \right).</math> As before, let <math>\langle y \mid x \rangle_H := \langle x, y \rangle_H</math> and <math>\langle y \mid x \rangle_Z := \langle x, y \rangle_Z.</math> 

Denote by
<math display=block>\begin{alignat}{4}
\Phi_H :\;&& H &&\;\to    \;& H^* \\[0.3ex]
          && g &&\;\mapsto\;& \langle \,g \mid \cdot\, \rangle_H \\
\end{alignat}
\quad \text{ and } \quad 
\begin{alignat}{4}
\Phi_Z :\;&& Z &&\;\to    \;& Z^* \\[0.3ex]
          && y &&\;\mapsto\;& \langle \,y  \mid \cdot\, \rangle_Z \\
\end{alignat}</math>
the usual bijective antilinear isometries that satisfy:
<math display=block>\left(\Phi_H g\right) h = \langle g\mid h \rangle_H \quad \text{ for all } g, h \in H \qquad \text{ and } \qquad \left(\Phi_Z y\right) z = \langle y \mid z \rangle_Z \quad \text{ for all } y, z \in Z.</math>

=== 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> 

=== 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}} 
<!-------------- START: Removed info --------------
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>
---------------- END:   Removed info -------------->
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.

===Descriptions of self-adjoint, normal, and unitary operators===

Assume <math>Z = H</math> and let <math>\Phi := \Phi_H = \Phi_Z.</math> 
Let <math>A : H \to H</math> be a continuous (that is, bounded) linear operator. 

Whether or not <math>A : H \to H</math> is [[Self-adjoint operator|self-adjoint]], [[Normal operator|normal]], or [[Unitary operator|unitary]] depends entirely on whether or not <math>A</math> satisfies certain defining conditions related to its adjoint, which was shown by ({{EquationNote|Adjoint-transpose}}) to essentially be just the transpose <math>{}^t A : H^* \to H^*.</math> 
Because the transpose of <math>A</math> is a map between continuous linear functionals, these defining conditions can consequently be re-expressed entirely in terms of linear functionals, as the remainder of subsection will now describe in detail. 
The linear functionals that are involved are the simplest possible continuous linear functionals on <math>H</math> that can be defined entirely in terms of <math>A,</math> the inner product <math>\langle \,\cdot\mid\cdot\, \rangle</math> on <math>H,</math> and some given vector <math>h \in H.</math> 
Specifically, these are <math>\left\langle A h\mid\cdot\, \right\rangle</math> and <math>\langle h\mid A (\cdot) \rangle</math><ref group=note name="ExplicitDefOfInnerProductOfTranspose" /> where 
<math display=block>\left\langle A h\mid\cdot\, \right\rangle = \Phi (A h) = (\Phi \circ A) h \quad \text{ and } \quad \langle h\mid A (\cdot) \rangle = \left({}^{t}A \circ \Phi\right) h.</math> 

'''Self-adjoint operators'''

{{See also|Self-adjoint operator|Hermitian matrix|Symmetric matrix}}

A continuous linear operator <math>A : H \to H</math> is called [[Self-adjoint operator|self-adjoint]] if it is equal to its own adjoint; that is, if <math>A = A^*.</math> Using ({{EquationNote|Adjoint-transpose}}), this happens if and only if:
<math display=block>\Phi \circ A = {}^t A \circ \Phi</math>
where this equality can be rewritten in the following two equivalent forms:
<math display=block>A = \Phi^{-1} \circ {}^t A \circ \Phi \quad \text{ or } \quad {}^{t}A = \Phi \circ A \circ \Phi^{-1}.</math>

Unraveling notation and definitions produces the following characterization of self-adjoint operators in terms of the aforementioned continuous linear functionals: <math>A</math> is self-adjoint if and only if for all <math>z \in H,</math> the linear functional <math>\langle z\mid A (\cdot) \rangle</math><ref group=note name="ExplicitDefOfInnerProductOfTranspose" /> is equal to the linear functional <math>\langle A z\mid\cdot\, \rangle</math>; that is, if and only if

{{NumBlk|:|<math>\langle z\mid A (\cdot) \rangle = \langle A z\mid \cdot\, \rangle \quad \text{ for all } z \in H</math>|{{EquationRef|Self-adjointness functionals}}|LnSty=1px dashed black}}

where if bra-ket notation is used, this is
<math display=block>\langle z \mid A ~=~ \langle A z \mid \quad \text{ for all } z \in H.</math> 

'''Normal operators'''

{{See also|Normal operator|Normal matrix}}

A continuous linear operator <math>A : H \to H</math> is called [[Normal operator|normal]] if <math>A A^* = A^* A,</math> which happens if and only if for all <math>z, h \in H,</math> 
<math display=block>\left\langle A A^* z\mid h \right\rangle = \left\langle A^* A z\mid h \right\rangle.</math> 

Using ({{EquationNote|Adjoint-transpose}}) and unraveling notation and definitions produces<ref group=proof name="NormalCharFunctionals" /> the following characterization of normal operators in terms of inner products of continuous linear functionals: <math>A</math> is a normal operator if and only if  

{{NumBlk|:|<math>\left\langle \,\langle A h \mid\cdot\, \rangle\mid\langle A z \mid\cdot\, \rangle\, \right\rangle_{H^*} ~=~ \left\langle \,\langle h | A(\cdot) \rangle\mid\langle z \mid A(\cdot) \rangle\, \right\rangle_{H^*} \quad \text{ for all } z, h \in H</math>|{{EquationRef|Normality functionals}}|LnSty=1px dashed black}}
where the left hand side is also equal to <math>\overline{\langle A h \mid A z \rangle}_H = \langle A z \mid A h \rangle_H.</math> 
The left hand side of this characterization involves ''only'' linear functionals of the form <math>\langle A h \mid\cdot\, \rangle</math> while the right hand side involves ''only'' linear functions of the form <math>\langle h \mid A(\cdot) \rangle</math> (defined as above<ref group=note name="ExplicitDefOfInnerProductOfTranspose" />). 
So in plain English, characterization ({{EquationNote|Normality functionals}}) says that an operator is ''normal'' when the inner product of any two linear functions of the first form is equal to the inner product of their second form (using the same vectors <math>z, h \in H</math> for both forms).
In other words, if it happens to be the case (and when <math>A</math> is injective or self-adjoint, it is) that the assignment of linear functionals <math>\langle A h \mid\cdot\, \rangle ~\mapsto~ \langle h | A(\cdot) \rangle</math> is well-defined (or alternatively, if <math>\langle h | A(\cdot) \rangle ~\mapsto~ \langle A h \mid\cdot\, \rangle</math> is well-defined) where <math>h</math> ranges over <math>H,</math> then <math>A</math> is a normal operator if and only if this assignment preserves the inner product on <math>H^*.</math> 

The fact that every self-adjoint bounded linear operator is normal follows readily by direct substitution of <math>A^* = A</math> into either side of <math>A^* A = A A^*.</math> 
This same fact also follows immediately from the direct substitution of the equalities ({{EquationNote|Self-adjointness functionals}}) into either side of ({{EquationNote|Normality functionals}}). 

Alternatively, for a complex Hilbert space, the continuous linear operator <math>A</math> is a normal operator if and only if <math>\|Az\| = \left\|A^* z\right\|</math> for every <math>z \in H,</math>{{sfn|Rudin|1991|pp=306-312}} which happens if and only if 
<math display=block>\|Az\|_H = \|\langle z\, | \,A(\cdot) \rangle\|_{H^*} \quad \text{ for every } z \in H.</math>

'''Unitary operators'''

{{See also|Unitary transformation|Unitary matrix}}

An invertible bounded linear operator <math>A : H \to H</math> is said to be [[Unitary operator|unitary]] if its inverse is its adjoint: <math>A^{-1} = A^*.</math> 
By using ({{EquationNote|Adjoint-transpose}}), this is seen to be equivalent to <math>\Phi \circ A^{-1} = {}^{t}A \circ \Phi.</math> 
Unraveling notation and definitions, it follows that <math>A</math> is unitary if and only if  
<math display=block>\langle A^{-1} z\mid\cdot\, \rangle = \langle z\mid A (\cdot) \rangle \quad \text{ for all } z \in H.</math>

The fact that a bounded invertible linear operator <math>A : H \to H</math> is unitary if and only if <math>A^* A = \operatorname{Id}_H</math> (or equivalently, <math>{}^t A \circ \Phi \circ A = \Phi</math>) produces another (well-known) characterization: an invertible bounded linear map <math>A</math> is unitary if and only if
<math display=block>\langle A z\mid A (\cdot)\, \rangle = \langle z\mid\cdot\, \rangle \quad \text{ for all } z \in H.</math>

Because <math>A : H \to H</math> is invertible (and so in particular a bijection), this is also true of the transpose <math>{}^t A : H^* \to H^*.</math> This fact also allows the vector <math>z \in H</math> in the above characterizations to be replaced with <math>A z</math> or <math>A^{-1} z,</math> thereby producing many more equalities. Similarly, <math>\,\cdot\,</math> can be replaced with <math>A(\cdot)</math> or <math>A^{-1}(\cdot).</math>

==See also==

* {{annotated link|Choquet theory}}
* {{annotated link|Covariance operator}}
* {{annotated link|Fundamental theorem of Hilbert spaces}}
* {{annotated link|Matrix coefficient}}

== Citations ==

{{reflist}}

== Notes ==

{{reflist|group=note|refs=
<ref group=note name="AntilinearIsometryDef">This means that for all vectors <math>y \in H:</math> (1) <math>\Phi : H \to H^*</math> is [[injective]]. (2) The [[Norm (mathematics)|norms]] of <math>y</math> and <math>\Phi(y)</math> are the same: <math>\|\Phi(y)\| = \|y\|.</math> (3) <math>\Phi</math> is an [[additive map]], meaning that <math>\Phi(x + y) = \Phi(x) + \Phi(y)</math> for all <math>x, y \in H.</math> (4) <math>\Phi</math> is [[conjugate homogeneous]]: <math>\Phi(s y) = \overline{s} \Phi(y)</math> for all scalars <math>s.</math> (5) <math>\Phi</math> is [[real homogeneous]]: <math>\Phi(r y) = r \Phi(y)</math> for all real numbers <math>r \in \R.</math></ref>

<ref group=note name="VectorSpaceStructureOnAffineHyperplanesInducedByDualSpace">This footnote explains how to define - using only <math>H</math>'s operations - addition and scalar multiplication of affine hyperplanes so that these operations correspond to addition and scalar multiplication of linear functionals. Let <math>H</math> be any vector space and let <math>H^{\#}</math> denote its [[algebraic dual space]]. Let <math>\mathcal{A} := \left\{ \varphi^{-1}(1) : \varphi \in H^{\#} \right\}</math> and let <math>\,\hat{\cdot}\,</math> and <math>\,\hat{+}\,</math> denote the (unique) vector space operations on <math>\mathcal{A}</math> that make the bijection <math>I : H^{\#} \to \mathcal{A}</math> defined by <math>\varphi \mapsto \varphi^{-1}(1)</math> into a [[vector space isomorphism]]. Note that <math>\varphi^{-1}(1) = \varnothing</math> if and only if <math>\varphi = 0,</math> so <math>\varnothing</math> is the additive identity of <math>\left(\mathcal{A}, \hat{+}, \hat{\cdot}\right)</math> (because this is true of <math>I^{-1}(\varnothing) = 0</math> in <math>H^{\#}</math> and <math>I</math> is a vector space isomorphism). For every <math>A \in \mathcal{A},</math> let <math>\ker A = H</math> if <math>A = \varnothing</math> and let <math>\ker A = A - A</math> otherwise; if <math>A = I(\varphi) = \varphi^{-1}(1)</math> then <math>\ker A = \ker \varphi</math> so this definition is consistent with the usual definition of the kernel of a linear functional. Say that <math>A, B \in \mathcal{A}</math> are {{em|parallel}} if <math>\ker A = \ker B,</math> where if <math>A</math> and <math>B</math> are not empty then this happens if and only if the linear functionals <math>I^{-1}(A)</math> and <math>I^{-1}(B)</math> are non-zero scalar multiples of each other. The vector space operations on the vector space of affine hyperplanes <math>\mathcal{A}</math> are now described in a way that involves {{em|only}} the vector space operations on <math>H</math>; this results in an interpretation of the vector space operations on the algebraic dual space <math>H^{\#}</math> that is entirely in terms of affine hyperplanes. Fix hyperplanes <math>A, B \in \mathcal{A}.</math> If <math>s</math> is a scalar then <math>s \hat{\cdot} A = \left\{ h \in H : s h \in A \right\}.</math> Describing the operation <math>A \hat{+} B</math> in terms of only the sets <math>A = \varphi^{-1}(1)</math> and <math>B = \psi^{-1}(1)</math> is more complicated because by definition, <math>A \hat{+} B = I(\varphi) \hat{+} I(\psi) := I(\varphi + \psi) = (\varphi + \psi)^{-1}(1).</math> If <math>A = \varnothing</math> (respectively, if <math>B = \varnothing</math>) then <math>A \hat{+} B</math> is equal to <math>B</math> (resp. is equal to <math>A</math>) so assume <math>A \neq \varnothing</math> and <math>B \neq \varnothing.</math> The hyperplanes <math>A</math> and <math>B</math> are parallel if and only if there exists some scalar <math>r</math> (necessarily non-0) such that <math>A = r B,</math> in which case <math>A \hat{+} B = \left\{ h \in H : (1 + r) h \in B \right\};</math> this can optionally be subdivided into two cases: if <math>r = -1</math> (which happens if and only if the linear functionals <math>I^{-1}(A)</math> and <math>I^{-1}(B)</math> are negatives of each) then <math>A \hat{+} B = \varnothing</math> while if <math>r \neq -1</math> then <math>A \hat{+} B = \frac{1}{1+r} B = \frac{r}{1+r} A.</math> Finally, assume now that <math>\ker A \neq \ker B.</math> Then <math>A \hat{+} B</math> is the unique affine hyperplane containing both <math>A \cap \ker B</math> and <math>B \cap \ker A</math> as subsets; explicitly, <math>\ker \left(A \hat{+} B\right) = \operatorname{span}\left(A \cap \ker B - B \cap \ker A\right)</math> and <math>A \hat{+} B = A \cap \ker B + \ker \left(A \hat{+} B\right) = B \cap \ker A + \ker \left(A \hat{+} B\right).</math> To see why this formula for <math>A \hat{+} B</math> should hold, consider <math>H := \R^3,</math> <math>A := \varphi^{-1}(1),</math> and <math>B := \psi^{-1}(1),</math> where <math>\varphi(x, y, z) := x</math> and <math>\psi(x, y, z) := x + y</math> (or alternatively, <math>\psi(x, y, z) := y</math>). Then by definition, <math>A \hat{+} B := (\varphi + \psi)^{-1}(1)</math> and <math>\ker \left(A \hat{+} B\right) := (\varphi + \psi)^{-1}(0).</math> Now <math>A \cap \ker B ~=~ \varphi^{-1}(1) \cap \psi^{-1}(0) ~\subseteq~ (\varphi + \psi)^{-1}(1)</math> is an affine subspace of [[codimension]] <math>2</math> in <math>H</math> (it is equal to a translation of the <math>z</math>-axis <math>\{(0, 0)\} \times \R</math>). The same is true of <math>B \cap \ker A.</math> Plotting an <math>x</math>-<math>y</math>-plane cross section (that is, setting <math>z = </math> constant) of the sets <math>\ker A, \ker B, A</math> and <math>B</math> (each of which will be plotted as a line), the set <math>(\varphi + \psi)^{-1}(1)</math> will then be plotted as the (unique) line passing through the <math>A \cap \ker B</math> and <math>B \cap \ker A</math> (which will be plotted as two distinct points) while <math>(\varphi + \psi)^{-1}(0) = \ker \left(A \hat{+} B\right)</math> will be plotted the line through the origin that is parallel to <math>A \hat{+} B = (\varphi + \psi)^{-1}(1).</math> The above formulas for <math>\ker \left(A \hat{+} B\right) := (\varphi + \psi)^{-1}(0)</math> and <math>A \hat{+} B := (\varphi + \psi)^{-1}(1)</math> follow naturally from the plot and they also hold in general.</ref>

<ref group=note name="ImportanceOfLocationOfRieszRep">If <math>\mathbb{F} = \R</math> then the inner product will be symmetric so it does not matter which coordinate of the inner product the element <math>y</math> is placed into because the same map will result. 
But if <math>\mathbb{F} = \Complex</math> then except for the constant <math>0</math> map, [[Antilinear map|antilinear functionals]] on <math>H</math> are completely distinct from [[linear functional]]s on <math>H,</math> which makes the coordinate that <math>y</math> is placed into is {{em|very}} important. 
For a non-zero <math>y \in H</math> to induce a {{em|linear}} functional (rather than an {{em|anti}}linear functional), <math>y</math> {{em|must}} be placed into the {{em|anti}}linear coordinate of the inner product. If it is incorrectly placed into the linear coordinate instead of the antilinear coordinate then the resulting map will be the antilinear map <math>h \mapsto \langle y , h \rangle = \langle h \mid y \rangle,</math> which is {{em|not}} a linear functional on <math>H</math> and so it will {{em|not}} be an element of the continuous dual space <math>H^*.</math></ref>

<ref group=note name="ExplicitDefOfInnerProductOfTranspose">The notation <math>\left\langle z \mid A(\cdot) \right\rangle_Z</math> denotes the continuous linear functional defined by <math>g \mapsto \left\langle z \mid A g \right\rangle_Z.</math></ref>
}}

'''Proofs'''

{{reflist|group=proof|refs=
<ref group=proof name="FormulaOrthoProjectionKernel">This is because <math>x_K = x - \frac{\left\langle x, f_\varphi \right\rangle}{\left\|f_\varphi\right\|^2} f_{\varphi}.</math> Now use <math>\left\| f_{\varphi} \right\|^2 = \|\varphi\|^2</math> and <math>\left\langle x, f_{\varphi} \right\rangle = \varphi(x)</math> and solve for <math>f_{\varphi}.</math> <math>\blacksquare</math></ref> 

<ref group=proof name="NormalCharFunctionals"><math>\left\langle A^* A z \mid h \right\rangle = \left\langle \,A z\mid A h\, \right\rangle_H = \left\langle \,\Phi A h\mid\Phi A z\, \right\rangle_{H^*}</math> where <math>\Phi A h := \left\langle A h\mid\cdot\, \right\rangle</math> and <math>\Phi A z := \left\langle A z \mid\cdot\, \right\rangle.</math> 
By definition of the adjoint, <math>\left\langle A^* h\mid A^* z\, \right\rangle = \left\langle h \mid A A^* z\, \right\rangle</math> so taking the complex conjugate of both sides proves that <math>\left\langle A A^* z\mid h \right\rangle = \left\langle A^* z\mid A^* h \right\rangle.</math> 
From <math>A^* = \Phi^{-1} \circ {}^{t}A \circ \Phi,</math> it follows that <math>\left\langle A A^* z\, | \,h \right\rangle_H 
= \left\langle A^* z\mid A^* h \right\rangle_H 
= \left\langle \Phi^{-1} \circ {}^{t}A \circ \Phi z\mid\Phi^{-1} \circ {}^t A \circ \Phi h \right\rangle_H 
= \left\langle \,{}^{t}A \circ \Phi h\mid{}^t A \circ \Phi z \right\rangle_{H^*}</math> where <math>\left({}^{t}A \circ \Phi\right) h = \langle h\, | \,A (\cdot) \rangle</math> and <math>\left({}^{t}A \circ \Phi\right) z = \langle z\, | \,A (\cdot) \rangle.</math> <math>\blacksquare</math></ref>
}}

==Bibliography==

* {{Bachman Narici Functional Analysis 2nd Edition}} <!--{{sfn|Bachman|Narici|2000|p=}}-->
* {{cite journal|last=Fréchet|first=M.|year=1907|title=Sur les ensembles de fonctions et les opérations linéaires|journal=[[Les Comptes rendus de l'Académie des sciences]]|volume=144|page=1414&ndash;1416|url=https://gallica.bnf.fr/ark:/12148/bpt6k3098j/f1414.image|language=fr}}
* [[P. Halmos]] ''Measure Theory'', D. van Nostrand and Co., 1950.
* P. Halmos, ''A Hilbert Space Problem Book'', Springer, New York 1982 ''(problem 3 contains version for vector spaces with coordinate systems)''.
* {{cite journal|last=Riesz|first=F.|year=1907|title=Sur une espèce de géométrie analytique des systèmes de fonctions sommables|journal=Comptes rendus de l'Académie des Sciences|volume=144|page=1409&ndash;1411|url=https://gallica.bnf.fr/ark:/12148/bpt6k3098j/f1409.image|language=fr}}
* {{cite journal|last=Riesz|first=F.|year=1909|title=Sur les opérations fonctionnelles linéaires|journal=Comptes rendus de l'Académie des Sciences|volume=149|page=974&ndash;977|url=https://gallica.bnf.fr/ark:/12148/bpt6k3103r/f976.image|language=fr}}

*{{citation | last=Roman | first=Stephen
 | title=Advanced Linear Algebra | edition=Third | series=[[Graduate Texts in Mathematics]] | publisher  = Springer | date=2008| pages= | isbn=978-0-387-72828-5 |author-link=Steven Roman}}
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin |1991|p=}} -->
* Walter Rudin, ''Real and Complex Analysis'', McGraw-Hill, 1966, {{isbn|0-07-100276-6}}.
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->

{{Functional analysis}}
{{Hilbert space}}

[[Category:Articles containing proofs]]
[[Category:Duality theories]]
[[Category:Hilbert spaces]]
[[Category:Integral representations]]
[[Category:Linear functionals]]
[[Category:Theorems in functional analysis]]