Riesz representation 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 is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism.

Preliminaries and notation[edit]

Let $H$ be a Hilbert space over a field $\mathbb {F} ,$ where $\mathbb {F}$ is either the real numbers $\mathbb {R}$ or the complex numbers $\mathbb {C} .$ If $\mathbb {F} =\mathbb {C}$ (resp. if $\mathbb {F} =\mathbb {R}$ ) then $H$ is called a complex Hilbert space (resp. a real Hilbert space). Every real Hilbert space can be extended to be a dense subset of a unique (up to 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 mathematicians and 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 $\mathbb {F} =\mathbb {R}$ ) 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 or complex Hilbert space.

Linear and antilinear maps[edit]

By definition, an antilinear map (also called a conjugate-linear map) $f:H\to Y$ is a map between vector spaces that is additive:

f(x+y)=f(x)+f(y)\quad {\text{ for all }}x,y\in H,

and antilinear (also called conjugate-linear or conjugate-homogeneous):

f(cx)={\overline {c}}f(x)\quad {\text{ for all }}x\in H{\text{ and all scalar }}c\in \mathbb {F} ,

where

{\overline {c}}

is the conjugate of the complex number

c=a+bi

, given by

{\overline {c}}=a-bi

.

In contrast, a map $f:H\to Y$ is linear if it is additive and homogeneous:

f(cx)=cf(x)\quad {\text{ for all }}x\in H\quad {\text{ and all scalars }}c\in \mathbb {F} .

Every constant $0$ map is always both linear and antilinear. If $\mathbb {F} =\mathbb {R}$ 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 if and only if it is bounded; the same is true of antilinear maps. The inverse of any antilinear (resp. linear) bijection is again an antilinear (resp. linear) bijection. The composition of two antilinear maps is a linear map.

Continuous dual and anti-dual spaces

A functional on $H$ is a function $H\to \mathbb {F}$ whose codomain is the underlying scalar field $\mathbb {F} .$ Denote by $H^{*}$ (resp. by ${\overline {H}}^{*})$ the set of all continuous linear (resp. continuous antilinear) functionals on $H,$ which is called the (continuous) dual space (resp. the (continuous) anti-dual space) of $H.$ ^[1] If $\mathbb {F} =\mathbb {R}$ then linear functionals on $H$ are the same as antilinear functionals and consequently, the same is true for such continuous maps: that is, $H^{*}={\overline {H}}^{*}.$

One-to-one correspondence between linear and antilinear functionals

Given any functional $f~:~H\to \mathbb {F} ,$ the conjugate of $f$ is the functional

{\begin{alignedat}{4}{\overline {f}}:\,&H&&\to \,&&\mathbb {F} \\&h&&\mapsto \,&&{\overline {f(h)}}.\\\end{alignedat}}

This assignment is most useful when $\mathbb {F} =\mathbb {C}$ because if $\mathbb {F} =\mathbb {R}$ then $f={\overline {f}}$ and the assignment $f\mapsto {\overline {f}}$ reduces down to the identity map.

The assignment $f\mapsto {\overline {f}}$ defines an antilinear bijective correspondence from the set of

all functionals (resp. all linear functionals, all continuous linear functionals

H^{*}

) on

H,

onto the set of

all functionals (resp. all antilinear functionals, all continuous antilinear functionals

{\overline {H}}^{*}

) on

H.

Mathematics vs. physics notations and definitions of inner product[edit]

The Hilbert space $H$ has an associated inner product $H\times H\to \mathbb {F}$ valued in $H$ 's underlying scalar field $\mathbb {F}$ that is linear in one coordinate and antilinear in the other (as described in detail below). If $H$ is a complex Hilbert space (meaning, if $\mathbb {F} =\mathbb {C}$ ), which is very often the case, then which coordinate is antilinear and which is linear becomes a very important technicality. However, if $\mathbb {F} =\mathbb {R}$ then the inner product is a symmetric map that is simultaneously linear in each coordinate (that is, bilinear) and antilinear in each coordinate. Consequently, the question of which coordinate is linear and which is antilinear is irrelevant for real Hilbert spaces.

Notation for the inner product

In mathematics, the inner product on a Hilbert space $H$ is often denoted by $\left\langle \cdot ,\cdot \right\rangle$ or $\left\langle \cdot ,\cdot \right\rangle _{H}$ while in physics, the bra–ket notation $\left\langle \cdot \mid \cdot \right\rangle$ or $\left\langle \cdot \mid \cdot \right\rangle _{H}$ is typically used instead. In this article, these two notations will be related by the equality:

\left\langle x,y\right\rangle :=\left\langle y\mid x\right\rangle \quad {\text{ for all }}x,y\in H.

Competing definitions of the inner product

The maps $\left\langle \cdot ,\cdot \right\rangle$ and $\left\langle \cdot \mid \cdot \right\rangle$ are assumed to have the following two properties:

The map $\left\langle \cdot ,\cdot \right\rangle$ $\left\langle \cdot ,\cdot \right\rangle$ is linear in its first coordinate; equivalently, the map $\left\langle \cdot \mid \cdot \right\rangle$ $\left\langle \cdot \mid \cdot \right\rangle$ is linear in its second coordinate. Explicitly, this means that for every fixed $y\in H,$ $y\in H,$ the map that is denoted by $\left\langle \,y\mid \cdot \,\right\rangle =\left\langle \,\cdot ,y\,\right\rangle :H\to \mathbb {F}$ $\left\langle \,y\mid \cdot \,\right\rangle =\left\langle \,\cdot ,y\,\right\rangle :H\to \mathbb {F}$ and defined by
$h\mapsto \left\langle \,y\mid h\,\right\rangle =\left\langle \,h,y\,\right\rangle \quad {\text{ for all }}h\in H$
$h\mapsto \left\langle \,y\mid h\,\right\rangle =\left\langle \,h,y\,\right\rangle \quad {\text{ for all }}h\in H$
is a linear functional on $H.$ $H.$
- In fact, this linear functional is continuous, so $\left\langle \,y\mid \cdot \,\right\rangle =\left\langle \,\cdot ,y\,\right\rangle \in H^{*}.$
The map $\left\langle \cdot ,\cdot \right\rangle$ $\left\langle \cdot ,\cdot \right\rangle$ is antilinear in its second coordinate; equivalently, the map $\left\langle \cdot \mid \cdot \right\rangle$ $\left\langle \cdot \mid \cdot \right\rangle$ is antilinear in its first coordinate. Explicitly, this means that for every fixed $y\in H,$ $y\in H,$ the map that is denoted by $\left\langle \,\cdot \mid y\,\right\rangle =\left\langle \,y,\cdot \,\right\rangle :H\to \mathbb {F}$ $\left\langle \,\cdot \mid y\,\right\rangle =\left\langle \,y,\cdot \,\right\rangle :H\to \mathbb {F}$ and defined by
$h\mapsto \left\langle \,h\mid y\,\right\rangle =\left\langle \,y,h\,\right\rangle \quad {\text{ for all }}h\in H$
$h\mapsto \left\langle \,h\mid y\,\right\rangle =\left\langle \,y,h\,\right\rangle \quad {\text{ for all }}h\in H$
is an antilinear functional on $H.$ $H.$
- In fact, this antilinear functional is continuous, so $\left\langle \,\cdot \mid y\,\right\rangle =\left\langle \,y,\cdot \,\right\rangle \in {\overline {H}}^{*}.$

In mathematics, the prevailing convention (i.e. the definition of an inner product) is that the inner product is linear in the first coordinate and antilinear in the other coordinate. In physics, the convention/definition is unfortunately the opposite, meaning that the inner product is linear in the second coordinate and antilinear in the other coordinate. This article will not choose one definition over the other. Instead, the assumptions made above make it so that the mathematics notation $\left\langle \cdot ,\cdot \right\rangle$ satisfies the mathematical convention/definition for the inner product (that is, linear in the first coordinate and antilinear in the other), while the physics bra–ket notation $\left\langle \cdot |\cdot \right\rangle$ satisfies the physics convention/definition for the inner product (that is, linear in the second coordinate and antilinear in the other). Consequently, the above two assumptions makes the notation used in each field consistent with that field's convention/definition for which coordinate is linear and which is antilinear.

Canonical norm and inner product on the dual space and anti-dual space[edit]

If $x=y$ then $\langle \,x\mid x\,\rangle =\langle \,x,x\,\rangle$ is a non-negative real number and the map

\|x\|:={\sqrt {\langle x,x\rangle }}={\sqrt {\langle x\mid x\rangle }}

defines a canonical norm on $H$ that makes $H$ into a normed space.^[1] As with all normed spaces, the (continuous) dual space $H^{*}$ carries a canonical norm, called the dual norm, that is defined by^[1]

\|f\|_{H^{*}}~:=~\sup _{\|x\|\leq 1,x\in H}|f(x)|\quad {\text{ for every }}f\in H^{*}.

The canonical norm on the (continuous) anti-dual space ${\overline {H}}^{*},$ denoted by $\|f\|_{{\overline {H}}^{*}},$ is defined by using this same equation:^[1]

\|f\|_{{\overline {H}}^{*}}~:=~\sup _{\|x\|\leq 1,x\in H}|f(x)|\quad {\text{ for every }}f\in {\overline {H}}^{*}.

This canonical norm on $H^{*}$ satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on $H^{*},$ which this article will denote by the notations

\left\langle f,g\right\rangle _{H^{*}}:=\left\langle g\mid f\right\rangle _{H^{*}},

where this inner product turns

H^{*}

into a Hilbert space. There are now two ways of defining a norm on

H^{*}:

the norm induced by this inner product (that is, the norm defined by

f\mapsto {\sqrt {\left\langle f,f\right\rangle _{H^{*}}}}

) 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

f\in H^{*}:

\sup _{\|x\|\leq 1,x\in H}|f(x)|=\|f\|_{H^{*}}~=~{\sqrt {\langle f,f\rangle _{H^{*}}}}~=~{\sqrt {\langle f\mid f\rangle _{H^{*}}}}.

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 $H^{*}.$

The same equations that were used above can also be used to define a norm and inner product on $H$ 's anti-dual space ${\overline {H}}^{*}.$ ^[1]

Canonical isometry between the dual and antidual

The complex conjugate ${\overline {f}}$ of a functional $f,$ which was defined above, satisfies

\|f\|_{H^{*}}~=~\left\|{\overline {f}}\right\|_{{\overline {H}}^{*}}\quad {\text{ and }}\quad \left\|{\overline {g}}\right\|_{H^{*}}~=~\|g\|_{{\overline {H}}^{*}}

for every

f\in H^{*}

and every

g\in {\overline {H}}^{*}.

This says exactly that the canonical antilinear bijection defined by

{\begin{alignedat}{4}\operatorname {Cong} :\;&&H^{*}&&\;\to \;&{\overline {H}}^{*}\\[0.3ex]&&f&&\;\mapsto \;&{\overline {f}}\\\end{alignedat}}

as well as its inverse

\operatorname {Cong} ^{-1}~:~{\overline {H}}^{*}\to H^{*}

are antilinear isometries and consequently also homeomorphisms. The inner products on the dual space

H^{*}

and the anti-dual space

{\overline {H}}^{*},

denoted respectively by

\langle \,\cdot \,,\,\cdot \,\rangle _{H^{*}}

and

\langle \,\cdot \,,\,\cdot \,\rangle _{{\overline {H}}^{*}},

are related by

\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^{*}

and

\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}}^{*}.

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

Riesz representation theorem[edit]

Two vectors $x$ and $y$ are orthogonal if $\langle x,y\rangle =0,$ which happens if and only if $\|y\|\leq \|y+sx\|$ for all scalars $s.$ ^[2] The orthogonal complement of a subset $X\subseteq H$ is

X^{\bot }:=\{\,y\in H:\langle y,x\rangle =0{\text{ for all }}x\in X\,\},

which is always a closed vector subspace of

H.

The Hilbert projection theorem guarantees that for any nonempty closed convex subset

C

of a Hilbert space there exists a unique vector

m\in C

such that

\|m\|=\inf _{c\in C}\|c\|;

that is,

m\in C

is the (unique) global minimum point of the function

C\to [0,\infty )

defined by

c\mapsto \|c\|.

Statement[edit]

Riesz representation theorem — Let $H$ be a Hilbert space whose inner product $\left\langle x,y\right\rangle$ is linear in its first argument and antilinear in its second argument and let $\langle y\mid x\rangle :=\langle x,y\rangle$ be the corresponding physics notation. For every continuous linear functional $\varphi \in H^{*},$ there exists a unique vector $f_{\varphi }\in H,$ called the Riesz representation of $\varphi ,$ such that^[3]

\varphi (x)=\left\langle x,f_{\varphi }\right\rangle =\left\langle f_{\varphi }\mid x\right\rangle \quad {\text{ for all }}x\in H.

Importantly for complex Hilbert spaces, $f_{\varphi }$ is always located in the antilinear coordinate of the inner product.^{[note 1]}

Furthermore, the length of the representation vector is equal to the norm of the functional:

\left\|f_{\varphi }\right\|_{H}=\|\varphi \|_{H^{*}},

and

f_{\varphi }

is the unique vector

f_{\varphi }\in \left(\ker \varphi \right)^{\bot }

with

\varphi \left(f_{\varphi }\right)=\|\varphi \|^{2}.

It is also the unique element of minimum norm in

C:=\varphi ^{-1}\left(\|\varphi \|^{2}\right)

; that is to say,

f_{\varphi }

is the unique element of

C

satisfying

\left\|f_{\varphi }\right\|=\inf _{c\in C}\|c\|.

Moreover, any non-zero

q\in (\ker \varphi )^{\bot }

can be written as

q=\left(\|q\|^{2}/\,{\overline {\varphi (q)}}\right)\ f_{\varphi }.

Corollary — The canonical map from $H$ into its dual $H^{*}$ ^[1] is the injective antilinear operator isometry^{[note 2]}^[1]

{\begin{alignedat}{4}\Phi :\;&&H&&\;\to \;&H^{*}\\[0.3ex]&&y&&\;\mapsto \;&\langle \,\cdot \,,y\rangle =\langle y|\,\cdot \,\rangle \\\end{alignedat}}

The Riesz representation theorem states that this map is surjective (and thus bijective) when

H

is complete and that its inverse is the bijective isometric antilinear isomorphism

{\begin{alignedat}{4}\Phi ^{-1}:\;&&H^{*}&&\;\to \;&H\\[0.3ex]&&\varphi &&\;\mapsto \;&f_{\varphi }\\\end{alignedat}}.

Consequently, every continuous linear functional on the Hilbert space

H

can be written uniquely in the form

\langle y\,|\,\cdot \,\rangle

^[1] where

\|\langle y\,|\cdot \rangle \|_{H^{*}}=\|y\|_{H}

for every

y\in H.

The assignment

y\mapsto \langle y,\cdot \rangle =\langle \cdot \,|\,y\rangle

can also be viewed as a bijective linear isometry

H\to {\overline {H}}^{*}

into the anti-dual space of

H,

^[1] which is the complex conjugate vector space of the continuous dual space

H^{*}.

The inner products on $H$ and $H^{*}$ are related by

\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

and similarly,

\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^{*}.

The set $C:=\varphi ^{-1}\left(\|\varphi \|^{2}\right)$ satisfies $C=f_{\varphi }+\ker \varphi$ and $C-f_{\varphi }=\ker \varphi$ so when $f_{\varphi }\neq 0$ then $C$ can be interpreted as being the affine hyperplane^{[note 3]} that is parallel to the vector subspace $\ker \varphi$ and contains $f_{\varphi }.$

For $y\in H,$ the physics notation for the functional $\Phi (y)\in H^{*}$ is the bra $\langle y|,$ where explicitly this means that $\langle y|:=\Phi (y),$ which complements the ket notation $|y\rangle$ defined by $|y\rangle :=y.$ 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 $\langle \psi \,|$ has a corresponding ket $|\,\psi \rangle ,$ and the latter is unique.

Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references).

Proof^[4]

Let $\mathbb {F}$ denote the underlying scalar field of $H.$

Proof of norm formula:

Fix $y\in H.$ Define $\Lambda :H\to \mathbb {F}$ by $\Lambda (z):=\langle \,y\,|\,z\,\rangle ,$ which is a linear functional on $H$ since $z$ is in the linear argument. By the Cauchy–Schwarz inequality,

|\Lambda (z)|=|\langle \,y\,|\,z\,\rangle |\leq \|y\|\|z\|

which shows that

\Lambda

is bounded (equivalently, continuous) and that

\|\Lambda \|\leq \|y\|.

It remains to show that

\|y\|\leq \|\Lambda \|.

By using

y

in place of

z,

it follows that

\|y\|^{2}=\langle \,y\,|\,y\,\rangle =\Lambda y=|\Lambda (y)|\leq \|\Lambda \|\|y\|

(the equality

\Lambda y=|\Lambda (y)|

holds because

\Lambda y=\|y\|^{2}\geq 0

is real and non-negative). Thus that

\|\Lambda \|=\|y\|.

\blacksquare

The proof above did not use the fact that $H$ is complete, which shows that the formula for the norm $\|\langle \,y\,|\,\cdot \,\rangle \|_{H^{*}}=\|y\|_{H}$ holds more generally for all inner product spaces.

Proof that a Riesz representation of $\varphi$ is unique:

Suppose $f,g\in H$ are such that $\varphi (z)=\langle \,f\,|\,z\,\rangle$ and $\varphi (z)=\langle \,g\,|\,z\,\rangle$ for all $z\in H.$ Then

\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

which shows that

\Lambda :=\langle \,f-g\,|\,\cdot \,\rangle

is the constant

0

linear functional. Consequently

0=\|\langle \,f-g\,|\,\cdot \,\rangle \|=\|f-g\|,

which implies that

f-g=0.

\blacksquare

Proof that a vector $f_{\varphi }$ representing $\varphi$ exists:

Let $K:=\ker \varphi :=\{m\in H:\varphi (m)=0\}.$ If $K=H$ (or equivalently, if $\varphi =0$ ) then taking $f_{\varphi }:=0$ completes the proof so assume that $K\neq H$ and $\varphi \neq 0.$ The continuity of $\varphi$ implies that $K$ is a closed subspace of $H$ (because $K=\varphi ^{-1}(\{0\})$ and $\{0\}$ is a closed subset of $\mathbb {F}$ ). Let

K^{\bot }:=\{v\in H~:~\langle \,v\,|\,k\,\rangle =0~{\text{ for all }}k\in K\}

denote the orthogonal complement of

K

in

H.

Because

K

is closed and

H

is a Hilbert space,^{[note 4]}

H

can be written as the direct sum

H=K\oplus K^{\bot }

^{[note 5]} (a proof of this is given in the article on the Hilbert projection theorem). Because

K\neq H,

there exists some non-zero

p\in K^{\bot }.

For any

h\in H,

\varphi [(\varphi h)p-(\varphi p)h]~=~\varphi [(\varphi h)p]-\varphi [(\varphi p)h]~=~(\varphi h)\varphi p-(\varphi p)\varphi h=0,

which shows that

(\varphi h)p-(\varphi p)h~\in ~\ker \varphi =K,

where now

p\in K^{\bot }

implies

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 .

Solving for

\varphi h

shows that

\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,

which proves that the vector

f_{\varphi }:={\frac {\overline {\varphi p}}{\|p\|^{2}}}p

satisfies

\varphi h=\langle \,f_{\varphi }\,|\,h\,\rangle {\text{ for every }}h\in H.

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

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

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

Observations[edit]

If $\varphi \in H^{*}$ then

\varphi \left(f_{\varphi }\right)=\left\langle f_{\varphi },f_{\varphi }\right\rangle =\left\|f_{\varphi }\right\|^{2}=\|\varphi \|^{2}.

So in particular,

\varphi \left(f_{\varphi }\right)\geq 0

is always real and furthermore,

\varphi \left(f_{\varphi }\right)=0

if and only if

f_{\varphi }=0

if and only if

\varphi =0.

Linear functionals as affine hyperplanes

A non-trivial continuous linear functional $\varphi$ is often interpreted geometrically by identifying it with the affine hyperplane $A:=\varphi ^{-1}(1)$ (the kernel $\ker \varphi =\varphi ^{-1}(0)$ is also often visualized alongside $A:=\varphi ^{-1}(1)$ although knowing $A$ is enough to reconstruct $\ker \varphi$ because if $A=\varnothing$ then $\ker \varphi =H$ and otherwise $\ker \varphi =A-A$ ). In particular, the norm of $\varphi$ should somehow be interpretable as the "norm of the hyperplane $A$ ". When $\varphi \neq 0$ then the Riesz representation theorem provides such an interpretation of $\|\varphi \|$ in terms of the affine hyperplane^{[note 3]} $A:=\varphi ^{-1}(1)$ as follows: using the notation from the theorem's statement, from $\|\varphi \|^{2}\neq 0$ it follows that $C:=\varphi ^{-1}\left(\|\varphi \|^{2}\right)=\|\varphi \|^{2}\varphi ^{-1}(1)=\|\varphi \|^{2}A$ and so $\|\varphi \|=\left\|f_{\varphi }\right\|=\inf _{c\in C}\|c\|$ implies $\|\varphi \|=\inf _{a\in A}\|\varphi \|^{2}\|a\|$ and thus $\|\varphi \|={\frac {1}{\inf _{a\in A}\|a\|}}.$ This can also be seen by applying the Hilbert projection theorem to $A$ and concluding that the global minimum point of the map $A\to [0,\infty )$ defined by $a\mapsto \|a\|$ is ${\frac {f_{\varphi }}{\|\varphi \|^{2}}}\in A.$ The formulas

{\frac {1}{\inf _{a\in A}\|a\|}}=\sup _{a\in A}{\frac {1}{\|a\|}}

provide the promised interpretation of the linear functional's norm

\|\varphi \|

entirely in terms of its associated affine hyperplane

A=\varphi ^{-1}(1)

(because with this formula, knowing only the set

A

is enough to describe the norm of its associated linear functional). Defining

{\frac {1}{\infty }}:=0,

the infimum formula

\|\varphi \|={\frac {1}{\inf _{a\in \varphi ^{-1}(1)}\|a\|}}

will also hold when

\varphi =0.

When the supremum is taken in

\mathbb {R}

(as is typically assumed), then the supremum of the empty set is

\sup \varnothing =-\infty

but if the supremum is taken in the non-negative reals

[0,\infty )

(which is the image/range of the norm

\|\,\cdot \,\|

when

\dim H>0

) then this supremum is instead

\sup \varnothing =0,

in which case the supremum formula

\|\varphi \|=\sup _{a\in \varphi ^{-1}(1)}{\frac {1}{\|a\|}}

will also hold when

\varphi =0

(although the atypical equality

\sup \varnothing =0

is usually unexpected and so risks causing confusion).

Constructions of the representing vector[edit]

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

f_{0}=0.

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

Orthogonal complement of kernel

If $\varphi \neq 0$ then for any $0\neq u\in (\ker \varphi )^{\bot },$

f_{\varphi }:={\frac {{\overline {\varphi (u)}}u}{\|u\|^{2}}}.

If $u\in (\ker \varphi )^{\bot }$ is a unit vector (meaning $\|u\|=1$ ) then

f_{\varphi }:={\overline {\varphi (u)}}u

(this is true even if

\varphi =0

because in this case

f_{\varphi }={\overline {\varphi (u)}}u={\overline {0}}u=0

). If

u

is a unit vector satisfying the above condition then the same is true of

-u,

which is also a unit vector in

(\ker \varphi )^{\bot }.

However,

{\overline {\varphi (-u)}}(-u)={\overline {\varphi (u)}}u=f_{\varphi }

so both these vectors result in the same

f_{\varphi }.

Orthogonal projection onto kernel

If $x\in H$ is such that $\varphi (x)\neq 0$ and if $x_{K}$ is the orthogonal projection of $x$ onto $\ker \varphi$ then^{[proof 1]}

f_{\varphi }={\frac {\|\varphi \|^{2}}{\varphi (x)}}\left(x-x_{K}\right).

Orthonormal basis

Given an orthonormal basis $\left\{e_{i}\right\}_{i\in I}$ of $H$ and a continuous linear functional $\varphi \in H^{*},$ the vector $f_{\varphi }\in H$ can be constructed uniquely by

f_{\varphi }=\sum _{i\in I}{\overline {\varphi \left(e_{i}\right)}}e_{i}

where all but at most countably many

\varphi \left(e_{i}\right)

will be equal to

0

and where the value of

f_{\varphi }

does not actually depend on choice of orthonormal basis (that is, using any other orthonormal basis for

H

will result in the same vector). If

y\in H

is written as

y=\sum _{i\in I}a_{i}e_{i}

then

\varphi (y)=\sum _{i\in I}\varphi \left(e_{i}\right)a_{i}=\langle f_{\varphi }|y\rangle

and

\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}.

If the orthonormal basis $\left\{e_{i}\right\}_{i\in I}=\left\{e_{i}\right\}_{i=1}^{\infty }$ is a sequence then this becomes

f_{\varphi }={\overline {\varphi \left(e_{1}\right)}}e_{1}+{\overline {\varphi \left(e_{2}\right)}}e_{2}+\cdots

and if

y\in H

is written as

y=\sum _{i\in I}a_{i}e_{i}=a_{1}e_{1}+a_{2}e_{2}+\cdots

then

\varphi (y)=\varphi \left(e_{1}\right)a_{1}+\varphi \left(e_{2}\right)a_{2}+\cdots =\langle f_{\varphi }|y\rangle .

Example in finite dimensions using matrix transformations[edit]

Consider the special case of $H=\mathbb {C} ^{n}$ (where $n>0$ is an integer) with the standard inner product

\langle z\mid w\rangle :={\overline {\,{\vec {z}}\,\,}}^{\operatorname {T} }{\vec {w}}\qquad {\text{ for all }}\;w,z\in H

where

w{\text{ and }}z

are represented as column matrices

{\vec {w}}:={\begin{bmatrix}w_{1}\\\vdots \\w_{n}\end{bmatrix}}

and

{\vec {z}}:={\begin{bmatrix}z_{1}\\\vdots \\z_{n}\end{bmatrix}}

with respect to the standard orthonormal basis

e_{1},\ldots ,e_{n}

on

H

(here,

e_{i}

is

1

at its

i

^th coordinate and

0

everywhere else; as usual,

H^{*}

will now be associated with the dual basis) and where

{\overline {\,{\vec {z}}\,}}^{\operatorname {T} }:=\left[{\overline {z_{1}}},\ldots ,{\overline {z_{n}}}\right]

denotes the conjugate transpose of

{\vec {z}}.

Let

\varphi \in H^{*}

be any linear functional and let

\varphi _{1},\ldots ,\varphi _{n}\in \mathbb {C}

be the unique scalars such that

\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,

where it can be shown that

\varphi _{i}=\varphi \left(e_{i}\right)

for all

i=1,\ldots ,n.

Then the Riesz representation of

\varphi

is the vector

f_{\varphi }~:=~{\overline {\varphi _{1}}}e_{1}+\cdots +{\overline {\varphi _{n}}}e_{n}~=~\left({\overline {\varphi _{1}}},\ldots ,{\overline {\varphi _{n}}}\right)\in H.

To see why, identify every vector

w=\left(w_{1},\ldots ,w_{n}\right)

in

H

with the column matrix

{\vec {w}}:={\begin{bmatrix}w_{1}\\\vdots \\w_{n}\end{bmatrix}}

so that

f_{\varphi }

is identified with

{\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}}.

As usual, also identify the linear functional

\varphi

with its transformation matrix, which is the row matrix

{\vec {\varphi }}:=\left[\varphi _{1},\ldots ,\varphi _{n}\right]

so that

{\vec {f_{\varphi }}}:={\overline {\,{\vec {\varphi }}\,\,}}^{\operatorname {T} }

and the function

\varphi

is the assignment

{\vec {w}}\mapsto {\vec {\varphi }}\,{\vec {w}},

where the right hand side is matrix multiplication. Then for all

w=\left(w_{1},\ldots ,w_{n}\right)\in H,

\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 ,

which shows that

f_{\varphi }

satisfies the defining condition of the Riesz representation of

\varphi .

The bijective antilinear isometry

\Phi :H\to H^{*}

defined in the corollary to the Riesz representation theorem is the assignment that sends

z=\left(z_{1},\ldots ,z_{n}\right)\in H

to the linear functional

\Phi (z)\in H^{*}

on

H

defined by

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},

where under the identification of vectors in

H

with column matrices and vector in

H^{*}

with row matrices,

\Phi

is just the assignment

{\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].

As described in the corollary,

\Phi

's inverse

\Phi ^{-1}:H^{*}\to H

is the antilinear isometry

\varphi \mapsto f_{\varphi },

which was just shown above to be:

\varphi ~\mapsto ~f_{\varphi }~:=~\left({\overline {\varphi \left(e_{1}\right)}},\ldots ,{\overline {\varphi \left(e_{n}\right)}}\right);

where in terms of matrices,

\Phi ^{-1}

is the assignment

{\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}}.

Thus in terms of matrices, each of

\Phi :H\to H^{*}

and

\Phi ^{-1}:H^{*}\to H

is just the operation of conjugate transposition

{\vec {v}}\mapsto {\overline {\,{\vec {v}}\,}}^{\operatorname {T} }

(although between different spaces of matrices: if

H

is identified with the space of all column (respectively, row) matrices then

H^{*}

is identified with the space of all row (respectively, column matrices).

This example used the standard inner product, which is the map $\langle z\mid w\rangle :={\overline {\,{\vec {z}}\,\,}}^{\operatorname {T} }{\vec {w}},$ but if a different inner product is used, such as $\langle z\mid w\rangle _{M}:={\overline {\,{\vec {z}}\,\,}}^{\operatorname {T} }\,M\,{\vec {w}}\,$ where $M$ is any 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[edit]

Assume that $H$ is a complex Hilbert space with inner product $\langle \,\cdot \mid \cdot \,\rangle .$ When the Hilbert space $H$

[1]

[2]

[3]

[note 1]

[note 2]

[note 3]

[4]

[note 4]

[note 5]

[proof 1]

Riesz representation theorem

Preliminaries and notation[edit]

Linear and antilinear maps[edit]

Mathematics vs. physics notations and definitions of inner product[edit]

Canonical norm and inner product on the dual space and anti-dual space[edit]

Riesz representation theorem[edit]

Statement[edit]

Observations[edit]

Constructions of the representing vector[edit]

Example in finite dimensions using matrix transformations[edit]

Relationship with the associated real Hilbert space[edit]

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