Theorem about the dual of a Hilbert space
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
Let
be a Hilbert space over a field
where
is either the real numbers
or the complex numbers
If
(resp. if
) then
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
) 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
By definition, an antilinear map (also called a conjugate-linear map)
is a map between vector spaces that is additive:
![{\displaystyle f(x+y)=f(x)+f(y)\quad {\text{ for all }}x,y\in H,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36ac07089eebf2c6d1299f6c93b7b654be913528)
and
antilinear (also called
conjugate-linear or
conjugate-homogeneous):
![{\displaystyle f(cx)={\overline {c}}f(x)\quad {\text{ for all }}x\in H{\text{ and all scalar }}c\in \mathbb {F} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/331b6363a37f7a97e4180d4bb981cb8a705f31c3)
where
![{\displaystyle {\overline {c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/930f590daba9bdddaa6d09c1569c64db5ea25707)
is the conjugate of the complex number
![{\displaystyle c=a+bi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a16bd13d5e61cf085ded0187d9adb099ac17ae0)
, given by
![{\displaystyle {\overline {c}}=a-bi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a47014cb016ae065cd04ff44f8661e362643784e)
.
In contrast, a map
is linear if it is additive and homogeneous:
![{\displaystyle f(cx)=cf(x)\quad {\text{ for all }}x\in H\quad {\text{ and all scalars }}c\in \mathbb {F} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b370b0d638baaf13efcac0e8576dc60a92a58ff)
Every constant
map is always both linear and antilinear. If
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
is a function
whose codomain is the underlying scalar field
Denote by
(resp. by
the set of all continuous linear (resp. continuous antilinear) functionals on
which is called the (continuous) dual space (resp. the (continuous) anti-dual space) of
If
then linear functionals on
are the same as antilinear functionals and consequently, the same is true for such continuous maps: that is,
One-to-one correspondence between linear and antilinear functionals
Given any functional
the conjugate of
is the functional
![{\displaystyle {\begin{alignedat}{4}{\overline {f}}:\,&H&&\to \,&&\mathbb {F} \\&h&&\mapsto \,&&{\overline {f(h)}}.\\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62dc42d76a5eef79cb9544d76a51953929df0551)
This assignment is most useful when
because if
then
and the assignment
reduces down to the identity map.
The assignment
defines an antilinear bijective correspondence from the set of
- all functionals (resp. all linear functionals, all continuous linear functionals
) on ![{\displaystyle H,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef601e1519093ba6c2944b945882c119f990e704)
onto the set of
- all functionals (resp. all antilinear functionals, all continuous antilinear functionals
) on ![{\displaystyle H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8933ae7244305ae7824aa18e077d1cf946e2ee9d)
Mathematics vs. physics notations and definitions of inner product
The Hilbert space
has an associated inner product
valued in
's underlying scalar field
that is linear in one coordinate and antilinear in the other (as described in detail below). If
is a complex Hilbert space (meaning, if
), which is very often the case, then which coordinate is antilinear and which is linear becomes a very important technicality. However, if
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
is often denoted by
or
while in physics, the bra–ket notation
or
is typically used instead. In this article, these two notations will be related by the equality:
![{\displaystyle \left\langle x,y\right\rangle :=\left\langle y\mid x\right\rangle \quad {\text{ for all }}x,y\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e9483053355d7597daff7c29c315d9873c8ef1b)
Competing definitions of the inner product
The maps
and
are assumed to have the following two properties:
- The map
is linear in its first coordinate; equivalently, the map
is linear in its second coordinate. Explicitly, this means that for every fixed
the map that is denoted by
and defined by ![{\displaystyle h\mapsto \left\langle \,y\mid h\,\right\rangle =\left\langle \,h,y\,\right\rangle \quad {\text{ for all }}h\in H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51635056198ee31be32a63aab568712cf0ab00f7)
is a linear functional on
- In fact, this linear functional is continuous, so
![{\displaystyle \left\langle \,y\mid \cdot \,\right\rangle =\left\langle \,\cdot ,y\,\right\rangle \in H^{*}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e83cd30e55cf80fe18f186815bc5e7611f2ac9a8)
- The map
is antilinear in its second coordinate; equivalently, the map
is antilinear in its first coordinate. Explicitly, this means that for every fixed
the map that is denoted by
and defined by ![{\displaystyle h\mapsto \left\langle \,h\mid y\,\right\rangle =\left\langle \,y,h\,\right\rangle \quad {\text{ for all }}h\in H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd9ff72f3fb75ee417ecb3f2b0540873d812f08)
is an antilinear functional on
- In fact, this antilinear functional is continuous, so
![{\displaystyle \left\langle \,\cdot \mid y\,\right\rangle =\left\langle \,y,\cdot \,\right\rangle \in {\overline {H}}^{*}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da0701f617cfe7d5292f175c31c1a2d076ab01c6)
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
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
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
If
then
is a non-negative real number and the map
![{\displaystyle \|x\|:={\sqrt {\langle x,x\rangle }}={\sqrt {\langle x\mid x\rangle }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71be19e3819dcd62e12b2c0f406df68f870b9ae0)
defines a canonical norm on
that makes
into a normed space. As with all normed spaces, the (continuous) dual space
carries a canonical norm, called the dual norm, that is defined by
![{\displaystyle \|f\|_{H^{*}}~:=~\sup _{\|x\|\leq 1,x\in H}|f(x)|\quad {\text{ for every }}f\in H^{*}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da49e533c9f53d4a151f96db91318b0a65e9aac1)
The canonical norm on the (continuous) anti-dual space
denoted by
is defined by using this same equation:
![{\displaystyle \|f\|_{{\overline {H}}^{*}}~:=~\sup _{\|x\|\leq 1,x\in H}|f(x)|\quad {\text{ for every }}f\in {\overline {H}}^{*}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/577c8add4fb17447799872bb03f1c53b191c1ff0)
This canonical norm on
satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on
which this article will denote by the notations
![{\displaystyle \left\langle f,g\right\rangle _{H^{*}}:=\left\langle g\mid f\right\rangle _{H^{*}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b076ecb7989a6209f3347ad74fa10e16afa716d)
where this inner product turns
![{\displaystyle H^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8ea7c513e82e824fda85563a940fbc2dc131fc4)
into a Hilbert space. There are now two ways of defining a norm on
![{\displaystyle H^{*}:}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0facd0948ccca31207379f968c0b234a2ca3c01b)
the norm induced by this inner product (that is, the norm defined by
![{\displaystyle f\mapsto {\sqrt {\left\langle f,f\right\rangle _{H^{*}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/061aa438e90e7813c88665603f35289a6b246f61)
) 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
![{\displaystyle \sup _{\|x\|\leq 1,x\in H}|f(x)|=\|f\|_{H^{*}}~=~{\sqrt {\langle f,f\rangle _{H^{*}}}}~=~{\sqrt {\langle f\mid f\rangle _{H^{*}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99b12e4054fd7d846442343959b00303eb38fb2b)
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
The same equations that were used above can also be used to define a norm and inner product on
's anti-dual space
Canonical isometry between the dual and antidual
The complex conjugate
of a functional
which was defined above, satisfies
![{\displaystyle \|f\|_{H^{*}}~=~\left\|{\overline {f}}\right\|_{{\overline {H}}^{*}}\quad {\text{ and }}\quad \left\|{\overline {g}}\right\|_{H^{*}}~=~\|g\|_{{\overline {H}}^{*}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/290e72844c16408ac162ed43bb7728d24445d6fa)
for every
![{\displaystyle f\in H^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea57d39afb89192be03138f0470ab51208d154f)
and every
![{\displaystyle g\in {\overline {H}}^{*}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bbbe3aede236582de2d127d77d540d0aecafda)
This says exactly that the canonical antilinear
bijection defined by
![{\displaystyle {\begin{alignedat}{4}\operatorname {Cong} :\;&&H^{*}&&\;\to \;&{\overline {H}}^{*}\\[0.3ex]&&f&&\;\mapsto \;&{\overline {f}}\\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb36a0b08c384fe66f94574963188c0a1dad6df3)
as well as its inverse
![{\displaystyle \operatorname {Cong} ^{-1}~:~{\overline {H}}^{*}\to H^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6a76933b828da7f210501e95a3685ef15db5d9)
are antilinear
isometries and consequently also
homeomorphisms. The inner products on the dual space
![{\displaystyle H^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8ea7c513e82e824fda85563a940fbc2dc131fc4)
and the anti-dual space
![{\displaystyle {\overline {H}}^{*},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87c474cc4c3102bbe5850e69297ee838131f5e18)
denoted respectively by
![{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle _{H^{*}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e11a9add1d250c66eb8447107fec9b99ffcc957f)
and
![{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle _{{\overline {H}}^{*}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa89d66ccc9979157fc30913011d053465397d0)
are related by
![{\displaystyle \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^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/934092d62102cd886d9603b9a80b0a2acf1aac57)
and
![{\displaystyle \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}}^{*}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1292b0ce34fd2d3e28ed362542755896c512127d)
If
then
and this canonical map
reduces down to the identity map.
Riesz representation theorem
Two vectors
and
are orthogonal if
which happens if and only if
for all scalars
The orthogonal complement of a subset
is
![{\displaystyle X^{\bot }:=\{\,y\in H:\langle y,x\rangle =0{\text{ for all }}x\in X\,\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe38c301e9fe21103060448b76f9704f9391c28)
which is always a
closed vector subspace of
![{\displaystyle H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8933ae7244305ae7824aa18e077d1cf946e2ee9d)
The
Hilbert projection theorem guarantees that for any
nonempty closed
convex subset ![{\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
of a
Hilbert space there exists a unique vector
![{\displaystyle m\in C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7522269d6d29898498443db0d387a52041b10b44)
such that
![{\displaystyle \|m\|=\inf _{c\in C}\|c\|;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a68f1f423496488bb7653e69a3b944c4dec32aed)
that is,
![{\displaystyle m\in C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7522269d6d29898498443db0d387a52041b10b44)
is the (unique)
global minimum point of the function
![{\displaystyle C\to [0,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5f468668be994cc9c22865293679218da9faae7)
defined by
Statement
Riesz representation theorem — Let
be a Hilbert space whose inner product
is linear in its first argument and antilinear in its second argument and let
be the corresponding physics notation. For every continuous linear functional
there exists a unique vector
called the Riesz representation of
such that[3]
![{\displaystyle \varphi (x)=\left\langle x,f_{\varphi }\right\rangle =\left\langle f_{\varphi }\mid x\right\rangle \quad {\text{ for all }}x\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49665e968ccae3cb9d25a3cdf543ec3cdf083493)
Importantly for complex Hilbert spaces,
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:
![{\displaystyle \left\|f_{\varphi }\right\|_{H}=\|\varphi \|_{H^{*}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78c5eeff4261b312c4b3e3ac218a00d057177185)
and
![{\displaystyle f_{\varphi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef316dd63d158ed103ecbe154ad1608b9304df4)
is the unique vector
![{\displaystyle f_{\varphi }\in \left(\ker \varphi \right)^{\bot }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4445d1ccfe6905fc43e63b35fcf7bf2be00cbf01)
with
![{\displaystyle \varphi \left(f_{\varphi }\right)=\|\varphi \|^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f3611a0b49bb075e2e030fe3583cf35813d4ba)
It is also the unique element of minimum norm in
![{\displaystyle C:=\varphi ^{-1}\left(\|\varphi \|^{2}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c32715e79a15febeb54f2693fc565fc0e9cee4ee)
; that is to say,
![{\displaystyle f_{\varphi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef316dd63d158ed103ecbe154ad1608b9304df4)
is the unique element of
![{\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
satisfying
![{\displaystyle \left\|f_{\varphi }\right\|=\inf _{c\in C}\|c\|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bff925a41146bb8f382a31f868ce836d7673d49)
Moreover, any non-zero
![{\displaystyle q\in (\ker \varphi )^{\bot }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/151c33e37e86a70828483549297700281df3f6d3)
can be written as
Corollary — The canonical map from
into its dual
is the injective antilinear operator isometry[note 2]
![{\displaystyle {\begin{alignedat}{4}\Phi :\;&&H&&\;\to \;&H^{*}\\[0.3ex]&&y&&\;\mapsto \;&\langle \,\cdot \,,y\rangle =\langle y|\,\cdot \,\rangle \\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e27af5a34637fbf79504e8525a73a7685d4c0d8)
The Riesz representation theorem states that this map is
surjective (and thus
bijective) when
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
is complete and that its inverse is the
bijective isometric antilinear isomorphism
![{\displaystyle {\begin{alignedat}{4}\Phi ^{-1}:\;&&H^{*}&&\;\to \;&H\\[0.3ex]&&\varphi &&\;\mapsto \;&f_{\varphi }\\\end{alignedat}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7275beb1d1c30539ac8568a3eb02920a56ae97c)
Consequently,
every continuous linear functional on the Hilbert space
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
can be written uniquely in the form
![{\displaystyle \langle y\,|\,\cdot \,\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b35f05059cafaedcd362e93ab70ebb09df68e4e9)
where
![{\displaystyle \|\langle y\,|\cdot \rangle \|_{H^{*}}=\|y\|_{H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69aeebca0d10da3c3cead1c04fd9a6386352e83e)
for every
![{\displaystyle y\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/299a560e74c8ebf12be3ad670052fb1f7fc60efc)
The assignment
![{\displaystyle y\mapsto \langle y,\cdot \rangle =\langle \cdot \,|\,y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/61a3d7df5c538257cda0b7498fafdd786ffd4c22)
can also be viewed as a bijective
linear isometry
![{\displaystyle H\to {\overline {H}}^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961606253af3d2bcb14c4c1a039d4485262da715)
into the
anti-dual space of
![{\displaystyle H,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef601e1519093ba6c2944b945882c119f990e704)
which is the
complex conjugate vector space of the
continuous dual space
The inner products on
and
are related by
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bacec317945dc87a82f52701983b8d6401b1451)
and similarly,
![{\displaystyle \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^{*}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/257da773c0f22c3643adfd4ec8b80fb2991c71dd)
The set
satisfies
and
so when
then
can be interpreted as being the affine hyperplane[note 3] that is parallel to the vector subspace
and contains
For
the physics notation for the functional
is the bra
where explicitly this means that
which complements the ket notation
defined by
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
has a corresponding ket
and the latter is unique.
Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references).
Proof |
Let denote the underlying scalar field of Proof of norm formula: Fix Define by which is a linear functional on since is in the linear argument. By the Cauchy–Schwarz inequality, ![{\displaystyle |\Lambda (z)|=|\langle \,y\,|\,z\,\rangle |\leq \|y\|\|z\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93cee89bdec3f4257ebc259a2522bcc6fca1138e) which shows that is bounded (equivalently, continuous) and that It remains to show that By using in place of it follows that ![{\displaystyle \|y\|^{2}=\langle \,y\,|\,y\,\rangle =\Lambda y=|\Lambda (y)|\leq \|\Lambda \|\|y\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfda8c5f2d8874c16707ff4ee6a89fb90ebc059e) (the equality holds because is real and non-negative). Thus that The proof above did not use the fact that is complete, which shows that the formula for the norm holds more generally for all inner product spaces. Proof that a Riesz representation of is unique: Suppose are such that and for all Then ![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c3152b46d469f759117bd0ad7ebd71b3bf4b50f) which shows that is the constant linear functional. Consequently which implies that Proof that a vector representing exists: Let If (or equivalently, if ) then taking completes the proof so assume that and The continuity of implies that is a closed subspace of (because and is a closed subset of ). Let ![{\displaystyle K^{\bot }:=\{v\in H~:~\langle \,v\,|\,k\,\rangle =0~{\text{ for all }}k\in K\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3701f5b454d3b50eae2db6fade27faace070c4c4) denote the orthogonal complement of in Because is closed and is a Hilbert space,[note 4] can be written as the direct sum [note 5] (a proof of this is given in the article on the Hilbert projection theorem). Because there exists some non-zero For any ![{\displaystyle \varphi [(\varphi h)p-(\varphi p)h]~=~\varphi [(\varphi h)p]-\varphi [(\varphi p)h]~=~(\varphi h)\varphi p-(\varphi p)\varphi h=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0bbf27289eb1fa2e22425b74ff78c3bf25c05e8) which shows that where now implies ![{\displaystyle 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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9198db45c8f8f15f348169ee68072a4e7a86a469) Solving for shows that ![{\displaystyle \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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2fe2dd4c46212a911c87a50cca08d40d647adb3) which proves that the vector satisfies Applying the norm formula that was proved above with shows that Also, the vector has norm and satisfies It can now be deduced that is -dimensional when Let be any non-zero vector. Replacing with in the proof above shows that the vector satisfies for every The uniqueness of the (non-zero) vector representing implies that which in turn implies that and Thus every vector in is a scalar multiple of The formulas for the inner products follow from the polarization identity. |
Observations
If
then
![{\displaystyle \varphi \left(f_{\varphi }\right)=\left\langle f_{\varphi },f_{\varphi }\right\rangle =\left\|f_{\varphi }\right\|^{2}=\|\varphi \|^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8cbef34e6ee60e440804849f9bc6b3515874bf2)
So in particular,
![{\displaystyle \varphi \left(f_{\varphi }\right)\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e26be7e9830a218397ba2af0ff814e5c130b0bf9)
is always real and furthermore,
![{\displaystyle \varphi \left(f_{\varphi }\right)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a60b701dd2a973d0ea4a9112100f6d86f388a24)
if and only if
![{\displaystyle f_{\varphi }=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1193882f4834ffb4ae3291a89ae50d4cb314d0fe)
if and only if
Linear functionals as affine hyperplanes
A non-trivial continuous linear functional
is often interpreted geometrically by identifying it with the affine hyperplane
(the kernel
is also often visualized alongside
although knowing
is enough to reconstruct
because if
then
and otherwise
). In particular, the norm of
should somehow be interpretable as the "norm of the hyperplane
". When
then the Riesz representation theorem provides such an interpretation of
in terms of the affine hyperplane[note 3]
as follows: using the notation from the theorem's statement, from
it follows that
and so
implies
and thus
This can also be seen by applying the Hilbert projection theorem to
and concluding that the global minimum point of the map
defined by
is
The formulas
![{\displaystyle {\frac {1}{\inf _{a\in A}\|a\|}}=\sup _{a\in A}{\frac {1}{\|a\|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0917290733311379f37d9e79779f25d1a37183)
provide the promised interpretation of the linear functional's norm
![{\displaystyle \|\varphi \|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35d07a928caade85c48ddd887d01b8c4ad1eca53)
entirely in terms of its associated affine hyperplane
![{\displaystyle A=\varphi ^{-1}(1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81743418ef70c6d0236f5aad6c7f14c2ac473464)
(because with this formula, knowing only the
set ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
is enough to describe the norm of its associated linear
functional). Defining
![{\displaystyle {\frac {1}{\infty }}:=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef7dedde29810f1b56049dac03ee79ad9901a0ca)
the
infimum formula
![{\displaystyle \|\varphi \|={\frac {1}{\inf _{a\in \varphi ^{-1}(1)}\|a\|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69e97cfd1ccd517085f1f7b6f788f614aab8edc2)
will also hold when
![{\displaystyle \varphi =0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28aee78b2dd5eb5250a41a10e0d4a8badadde92d)
When the supremum is taken in
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
(as is typically assumed), then the supremum of the empty set is
![{\displaystyle \sup \varnothing =-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2b29333e7e20077a76d153037ab27bdc6dad3f)
but if the supremum is taken in the non-negative reals
![{\displaystyle [0,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc2d914c2df66bc0f7893bfb8da36766650fe47)
(which is the
image/range of the norm
![{\displaystyle \|\,\cdot \,\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4191bf634388433a0692daefa7e5b93ba5422ece)
when
![{\displaystyle \dim H>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a030252adb42cb399d8fe235e4c45a14c16992cc)
) then this supremum is instead
![{\displaystyle \sup \varnothing =0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e58b8f841fd0d074f20f49cefbd3d6614cc61348)
in which case the supremum formula
![{\displaystyle \|\varphi \|=\sup _{a\in \varphi ^{-1}(1)}{\frac {1}{\|a\|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc7f5a03abff8ff65209e36606b4341cc274671)
will also hold when
![{\displaystyle \varphi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/192287b02f5764a18fe39f37b8199d72000aa220)
(although the atypical equality
![{\displaystyle \sup \varnothing =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c2ff3ef945e86ff08ba76d458899286dc1dda3b)
is usually unexpected and so risks causing confusion).
Constructions of the representing vector
Using the notation from the theorem above, several ways of constructing
from
are now described. If
then
; in other words,
![{\displaystyle f_{0}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2cae649a69248c2eb0ea0f94455c09f001d66bb)
This special case of
is henceforth assumed to be known, which is why some of the constructions given below start by assuming
Orthogonal complement of kernel
If
then for any
![{\displaystyle f_{\varphi }:={\frac {{\overline {\varphi (u)}}u}{\|u\|^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5acbc6cd897ed48894b3acab1cac92205df5fcb)
If
is a unit vector (meaning
) then
![{\displaystyle f_{\varphi }:={\overline {\varphi (u)}}u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3415f65e6d1679808eaa4973a8833319a40d49c4)
(this is true even if
![{\displaystyle \varphi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/192287b02f5764a18fe39f37b8199d72000aa220)
because in this case
![{\displaystyle f_{\varphi }={\overline {\varphi (u)}}u={\overline {0}}u=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1dce19974952fe2e80c60f6cb611af61e4beb2b)
). If
![{\displaystyle u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8)
is a unit vector satisfying the above condition then the same is true of
![{\displaystyle -u,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db225e795ccc921743185254063407537ee5e2d7)
which is also a unit vector in
![{\displaystyle (\ker \varphi )^{\bot }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f4a3e153dc56813a131e0e9fe697120b3b110d)
However,
![{\displaystyle {\overline {\varphi (-u)}}(-u)={\overline {\varphi (u)}}u=f_{\varphi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e52951be67b414d6946b0a66f417600708b56a77)
so both these vectors result in the same
Orthogonal projection onto kernel
If
is such that
and if
is the orthogonal projection of
onto
then[proof 1]
![{\displaystyle f_{\varphi }={\frac {\|\varphi \|^{2}}{\varphi (x)}}\left(x-x_{K}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c652a57e17e7680965aeadeb43225d750abb3593)
Orthonormal basis
Given an orthonormal basis
of
and a continuous linear functional
the vector
can be constructed uniquely by
![{\displaystyle f_{\varphi }=\sum _{i\in I}{\overline {\varphi \left(e_{i}\right)}}e_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dbedb6f2024ff01ca0a7977402e3868ff2cd794)
where all but at most countably many
![{\displaystyle \varphi \left(e_{i}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/588906548e95f04f1f87260165f8e1dd74e6070c)
will be equal to
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
and where the value of
![{\displaystyle f_{\varphi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef316dd63d158ed103ecbe154ad1608b9304df4)
does not actually depend on choice of orthonormal basis (that is, using any other orthonormal basis for
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
will result in the same vector). If
![{\displaystyle y\in H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3b8cfbeab2ec976c41ccc1573c57ef839d6c07)
is written as
![{\displaystyle y=\sum _{i\in I}a_{i}e_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45ca178ef56ae5cb8e191a1aadcea35cf623f070)
then
![{\displaystyle \varphi (y)=\sum _{i\in I}\varphi \left(e_{i}\right)a_{i}=\langle f_{\varphi }|y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/443a097c22fbabb5681036868c01f303dec6152a)
and
![{\displaystyle \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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f5eea85b897d81b20814ec9b7a719e97fc028a)
If the orthonormal basis
is a sequence then this becomes
![{\displaystyle f_{\varphi }={\overline {\varphi \left(e_{1}\right)}}e_{1}+{\overline {\varphi \left(e_{2}\right)}}e_{2}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e05e36e23cfa8ece5c4e3a8519415e98b5e6beb0)
and if
![{\displaystyle y\in H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3b8cfbeab2ec976c41ccc1573c57ef839d6c07)
is written as
![{\displaystyle y=\sum _{i\in I}a_{i}e_{i}=a_{1}e_{1}+a_{2}e_{2}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/75d7477ec97ccecd5b718de7ba398facf37108bf)
then
![{\displaystyle \varphi (y)=\varphi \left(e_{1}\right)a_{1}+\varphi \left(e_{2}\right)a_{2}+\cdots =\langle f_{\varphi }|y\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97e5d8ff4cf0f4ecb40f96267add5ef5d8d7b28e)
Example in finite dimensions using matrix transformations
Consider the special case of
(where
is an integer) with the standard inner product
![{\displaystyle \langle z\mid w\rangle :={\overline {\,{\vec {z}}\,\,}}^{\operatorname {T} }{\vec {w}}\qquad {\text{ for all }}\;w,z\in H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0430f2726b276282e2267f95b9b5c61d7e5bcadc)
where
![{\displaystyle w{\text{ and }}z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c4877e54fbc4d8651263f350c8fd86fbbb13e2)
are represented as
column matrices ![{\displaystyle {\vec {w}}:={\begin{bmatrix}w_{1}\\\vdots \\w_{n}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f62109338132e5267978b23d66aec66970a0ce)
and
![{\displaystyle {\vec {z}}:={\begin{bmatrix}z_{1}\\\vdots \\z_{n}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb79babfbbb70f9f800fecaccf16d03430aed70d)
with respect to the standard orthonormal basis
![{\displaystyle e_{1},\ldots ,e_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c60c38b7e2450d62e9dc496b89f8e5c96c77cecf)
on
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
(here,
![{\displaystyle e_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdc3a9cb1583d3204eff8918b558c293e0d2cf3)
is
![{\displaystyle 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
at its
th coordinate and
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
everywhere else; as usual,
![{\displaystyle H^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8ea7c513e82e824fda85563a940fbc2dc131fc4)
will now be associated with the
dual basis) and where
![{\displaystyle {\overline {\,{\vec {z}}\,}}^{\operatorname {T} }:=\left[{\overline {z_{1}}},\ldots ,{\overline {z_{n}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22bd4da85e46f1fe1f5a7e0b6abdb1300f468fb4)
denotes the
conjugate transpose of
![{\displaystyle {\vec {z}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8186822da92808b8d6c15e3494c77aa7de1699)
Let
![{\displaystyle \varphi \in H^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16a8e4288ce7091fcd1eb3a14ab1b65b9ad5668f)
be any linear functional and let
![{\displaystyle \varphi _{1},\ldots ,\varphi _{n}\in \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/732bff8fd34231ebb2f67569ae0f6db325e97cff)
be the unique scalars such that
![{\displaystyle \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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27cc11f787c6221ad3b1c1a995036161ea4d2069)
where it can be shown that
![{\displaystyle \varphi _{i}=\varphi \left(e_{i}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78c365fb3b48858b962e93541bce868b3143f9ac)
for all
![{\displaystyle i=1,\ldots ,n.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df1fdaf4311ca9e5826a22477bcd28ce4b042fcf)
Then the Riesz representation of
![{\displaystyle \varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
is the vector
![{\displaystyle f_{\varphi }~:=~{\overline {\varphi _{1}}}e_{1}+\cdots +{\overline {\varphi _{n}}}e_{n}~=~\left({\overline {\varphi _{1}}},\ldots ,{\overline {\varphi _{n}}}\right)\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8a58a73b72842b32cc9c6639f9624482c44c12)
To see why, identify every vector
![{\displaystyle w=\left(w_{1},\ldots ,w_{n}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7af33489ccbfccd49f244f3bb0be7dd258bed3a)
in
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
with the column matrix
![{\displaystyle {\vec {w}}:={\begin{bmatrix}w_{1}\\\vdots \\w_{n}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f62109338132e5267978b23d66aec66970a0ce)
so that
![{\displaystyle f_{\varphi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef316dd63d158ed103ecbe154ad1608b9304df4)
is identified with
![{\displaystyle {\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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efa07fdfb16d12ee8af92fa9d625d3cbc07456b5)
As usual, also identify the linear functional
![{\displaystyle \varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
with its
transformation matrix, which is the
row matrix ![{\displaystyle {\vec {\varphi }}:=\left[\varphi _{1},\ldots ,\varphi _{n}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/147d47a10aa6e42bb6e9d8d0014bcdb1ad0c6e84)
so that
![{\displaystyle {\vec {f_{\varphi }}}:={\overline {\,{\vec {\varphi }}\,\,}}^{\operatorname {T} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39103c0a1b57073e086023ca28f69da65130761f)
and the function
![{\displaystyle \varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
is the assignment
![{\displaystyle {\vec {w}}\mapsto {\vec {\varphi }}\,{\vec {w}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c26c1bf889885fc40ed0d902a8f2c72c40ec74)
where the right hand side is
matrix multiplication. Then for all
![{\displaystyle \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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99680b4b0812733c07934b441cb0be264bc4468b)
which shows that
![{\displaystyle f_{\varphi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef316dd63d158ed103ecbe154ad1608b9304df4)
satisfies the defining condition of the Riesz representation of
![{\displaystyle \varphi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b6c90c1e9984232aed2d530ac2fb2660ea000a)
The bijective antilinear isometry
![{\displaystyle \Phi :H\to H^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d95c9800df4158a7dbf856e60afb527886bac6bd)
defined in the corollary to the Riesz representation theorem is the assignment that sends
![{\displaystyle z=\left(z_{1},\ldots ,z_{n}\right)\in H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f3e9cc9baef59e1437dfae236e957db2836aacd)
to the linear functional
![{\displaystyle \Phi (z)\in H^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/548597493b3a355400291f8fa9837af0f72fc086)
on
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
defined by
![{\displaystyle 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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/645990523cbe451e5cfd5ac16d830ece9d445a7a)
where under the identification of vectors in
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
with column matrices and vector in
![{\displaystyle H^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8ea7c513e82e824fda85563a940fbc2dc131fc4)
with row matrices,
![{\displaystyle \Phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24)
is just the assignment
![{\displaystyle {\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a23fdc50c15c7972194c18a3ff989ce4fe8f221)
As described in the corollary,
![{\displaystyle \Phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24)
's inverse
![{\displaystyle \Phi ^{-1}:H^{*}\to H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1321f38c2fde8920e1ad12bc8a093e7c4cec88)
is the antilinear isometry
![{\displaystyle \varphi \mapsto f_{\varphi },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79cb0545803b6671bac195b5ae77d223c10bc766)
which was just shown above to be:
![{\displaystyle \varphi ~\mapsto ~f_{\varphi }~:=~\left({\overline {\varphi \left(e_{1}\right)}},\ldots ,{\overline {\varphi \left(e_{n}\right)}}\right);}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7406b5ef84bc6b2e81bf0a5d76078646d9ce6956)
where in terms of matrices,
![{\displaystyle \Phi ^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab22c7cf7f1a54d85993e0257a93f28eae546df8)
is the assignment
![{\displaystyle {\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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01fd9a7811a1f9ca97c5511f5dac42e49159ecff)
Thus in terms of matrices, each of
![{\displaystyle \Phi :H\to H^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d95c9800df4158a7dbf856e60afb527886bac6bd)
and
![{\displaystyle \Phi ^{-1}:H^{*}\to H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1321f38c2fde8920e1ad12bc8a093e7c4cec88)
is just the operation of
conjugate transposition ![{\displaystyle {\vec {v}}\mapsto {\overline {\,{\vec {v}}\,}}^{\operatorname {T} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19f0e9f65a3532e0e120d874128509dce5eec6c1)
(although between different spaces of matrices: if
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
is identified with the space of all column (respectively, row) matrices then
![{\displaystyle H^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8ea7c513e82e824fda85563a940fbc2dc131fc4)
is identified with the space of all row (respectively, column matrices).
This example used the standard inner product, which is the map
but if a different inner product is used, such as
where
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
Assume that
is a complex Hilbert space with inner product
When the Hilbert space
is reinterpreted as a real Hilbert space then it will be denoted by
where the (real) inner-product on
is the real part of
's inner product; that is:
![{\displaystyle \langle x,y\rangle _{\mathbb {R} }:=\operatorname {re} \langle x,y\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/679a2781da3e699e39843b3a24b1ca515a498bce)
The norm on
induced by
is equal to the original norm on
and the continuous dual space of
is the set of all real-valued bounded
-linear functionals on
(see the article about the polarization identity for additional details about this relationship). Let
and
denote the real and imaginary parts of a linear functional
so that
The formula expressing a linear functional in terms of its real part is
![{\displaystyle \psi (h)=\psi _{\mathbb {R} }(h)-i\psi _{\mathbb {R} }(ih)\quad {\text{ for }}h\in H,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0cab90d73149cf16749f2afec8c15015182e8e)
where
![{\displaystyle \psi _{i}(h)=-i\psi _{\mathbb {R} }(ih)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa3b11edb27c559976419e7b1396f6ab30cdddf)
for all
![{\displaystyle h\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d10c356f8b2cea62359891c45c61a5670d8ade85)
It follows that
![{\displaystyle \ker \psi _{\mathbb {R} }=\psi ^{-1}(i\mathbb {R} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff44d7567cca1d00aaf3568bd98045cdf0233a2b)
and that
![{\displaystyle \psi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7eca9b06e607571768c9b79cb231a9997308c4e6)
if and only if
![{\displaystyle \psi _{\mathbb {R} }=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/841cd4977abaa221067db5ae0f649df8c3c1879d)
It can also be shown that
![{\displaystyle \|\psi \|=\left\|\psi _{\mathbb {R} }\right\|=\left\|\psi _{i}\right\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1409cdb41112a6adbf6d4dfe5df2ae0b545d6c3)
where
![{\displaystyle \left\|\psi _{\mathbb {R} }\right\|:=\sup _{\|h\|\leq 1}\left|\psi _{\mathbb {R} }(h)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/383c8778f86559db9fa605f6df8abd19b0f04411)
and
![{\displaystyle \left\|\psi _{i}\right\|:=\sup _{\|h\|\leq 1}\left|\psi _{i}(h)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d325a7e2a11fdcce1d8eef66c26b8c24c8fba623)
are the usual
operator norms. In particular, a linear functional
![{\displaystyle \psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a)
is bounded if and only if its real part
![{\displaystyle \psi _{\mathbb {R} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac1a1ad4315fc5f3bc27c92ab0e2287f637822a)
is bounded.
Representing a functional and its real part
The Riesz representation of a continuous linear function
on a complex Hilbert space is equal to the Riesz representation of its real part
on its associated real Hilbert space.
Explicitly, let
and as above, let
be the Riesz representation of
obtained in
so it is the unique vector that satisfies
for all
The real part of
is a continuous real linear functional on
and so the Riesz representation theorem may be applied to
and the associated real Hilbert space
to produce its Riesz representation, which will be denoted by
That is,
is the unique vector in
that satisfies
for all
The conclusion is
This follows from the main theorem because
and if
then
![{\displaystyle \left\langle f_{\varphi }\mid x\right\rangle _{\mathbb {R} }=\operatorname {re} \left\langle f_{\varphi }\mid x\right\rangle =\operatorname {re} \varphi (x)=\varphi _{\mathbb {R} }(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2383fa4caccf19657f8521508b6ed6c233226ef7)
and consequently, if
![{\displaystyle m\in \ker \varphi _{\mathbb {R} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5218c4083852ea74d3c01321951bb08e4e9ec1a)
then
![{\displaystyle \left\langle f_{\varphi }\mid m\right\rangle _{\mathbb {R} }=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de2a428eafbcdfcecf262bbf3721df85ad7318c8)
which shows that
![{\displaystyle f_{\varphi }\in (\ker \varphi _{\mathbb {R} })^{\perp _{\mathbb {R} }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e39fdcdf29b9d093804156eaab131efad6d79275)
Moreover,
![{\displaystyle \varphi (f_{\varphi })=\|\varphi \|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/538689ad4fee91e8cdcfd4bdc3ec510cfade09d0)
being a real number implies that
![{\displaystyle \varphi _{\mathbb {R} }(f_{\varphi })=\operatorname {re} \varphi (f_{\varphi })=\|\varphi \|^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff713ed129cf2e23fbfdf236c513e1816a374f4b)
In other words, in the theorem and constructions above, if
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
is replaced with its real Hilbert space counterpart
![{\displaystyle H_{\mathbb {R} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4594f07af31202e24d1c21ae0c8bb424789df37)
and if
![{\displaystyle \varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
is replaced with
![{\displaystyle \operatorname {re} \varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccfe0cc32db1661988c97510ba0b63c1acc23c39)
then
![{\displaystyle f_{\varphi }=f_{\operatorname {re} \varphi }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24f5cd71db36ba02f27f0cdad053e29e734fcf67)
This means that vector
![{\displaystyle f_{\varphi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef316dd63d158ed103ecbe154ad1608b9304df4)
obtained by using
![{\displaystyle \left(H_{\mathbb {R} },\langle ,\cdot ,\cdot \rangle _{\mathbb {R} }\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb4d98b4683e673fbd86e21b80a5bc313029c9d8)
and the real linear functional
![{\displaystyle \operatorname {re} \varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccfe0cc32db1661988c97510ba0b63c1acc23c39)
is the equal to the vector obtained by using the origin complex Hilbert space
![{\displaystyle \left(H,\left\langle ,\cdot ,\cdot \right\rangle \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b75a0d1f8dd1b9caae82981bf3c22bdb23e7b2e6)
and original complex linear functional
![{\displaystyle \varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
(with identical norm values as well).
Furthermore, if
then
is perpendicular to
with respect to
where the kernel of
is be a proper subspace of the kernel of its real part
Assume now that
Then
because
and
is a proper subset of
The vector subspace
has real codimension
in
while
has real codimension
in
and
That is,
is perpendicular to
with respect to
Canonical injections into the dual and anti-dual
Induced linear map into anti-dual
The map defined by placing
into the linear coordinate of the inner product and letting the variable
vary over the antilinear coordinate results in an antilinear functional:
![{\displaystyle \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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ed0e8329aa8d118445437e83d9cfc73d048d01)
This map is an element of
which is the continuous anti-dual space of
The canonical map from
into its anti-dual
is the linear operator
![{\displaystyle {\begin{alignedat}{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{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62715e4539051f8805373fa3099e08644e8577df)
which is also an
injective isometry. The
Fundamental theorem of Hilbert spaces, which is related to Riesz representation theorem, states that this map is surjective (and thus
bijective). Consequently, every antilinear functional on
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
can be written (uniquely) in this form.
If
is the canonical antilinear bijective isometry
that was defined above, then the following equality holds:
![{\displaystyle \operatorname {Cong} ~\circ ~\operatorname {In} _{H}^{H^{*}}~=~\operatorname {In} _{H}^{{\overline {H}}^{*}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae2746ea50478b932f94c978cc5014b8df6d24e)
Extending the bra–ket notation to bras and kets
Let
be a Hilbert space and as before, let
Let
![{\displaystyle {\begin{alignedat}{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{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/028e18a1c1d3520d98292b413c114482174776d1)
which is a bijective antilinear isometry that satisfies
![{\displaystyle (\Phi h)g=\langle h\mid g\rangle _{H}=\langle g,h\rangle _{H}\quad {\text{ for all }}g,h\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/571125aa27b299116daf60cde9f2c32532740658)
Bras
Given a vector
let
denote the continuous linear functional
; that is,
![{\displaystyle \langle h\,|~:=~\Phi h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/398557389a3d83f12a45a577af7e5774673e13ad)
so that this functional
![{\displaystyle \langle h\,|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d5bb5fe1f2d6cd1dd00d5ebe383371e3b03869)
is defined by
![{\displaystyle g\mapsto \left\langle \,h\mid g\,\right\rangle _{H}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e87dc8a2a8bed62b9a41fbe588b5d26f6e6274)
This map was denoted by
![{\displaystyle \left\langle h\mid \cdot \,\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a51e573c781614d6ca506387f503922ca2e590db)
earlier in this article.
The assignment
is just the isometric antilinear isomorphism
which is why
holds for all
and all scalars
The result of plugging some given
into the functional
is the scalar
which may be denoted by
[note 6]
Bra of a linear functional
Given a continuous linear functional
let
denote the vector
; that is,
![{\displaystyle \langle \psi \mid ~:=~\Phi ^{-1}\psi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93c063a0368b76fc871572a2f9c7a4218ed89299)
The assignment
is just the isometric antilinear isomorphism
which is why
holds for all
and all scalars
The defining condition of the vector
is the technically correct but unsightly equality
![{\displaystyle \left\langle \,\langle \psi \mid \,\mid g\right\rangle _{H}~=~\psi g\quad {\text{ for all }}g\in H,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70dc8cb0ffb5a2b381bdc8c163fe429305512964)
which is why the notation
![{\displaystyle \left\langle \psi \mid g\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8e130017fd5c86f448df8f044ad7e2c89b95a12)
is used in place of
![{\displaystyle \left\langle \,\langle \psi \mid \,\mid g\right\rangle _{H}=\left\langle g,\,\langle \psi \mid \right\rangle _{H}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ea2661b3b9c3a3726932fbf3097358197ead36)
With this notation, the defining condition becomes
![{\displaystyle \left\langle \psi \mid g\right\rangle ~=~\psi g\quad {\text{ for all }}g\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ec40a69c638ac7b87873f0299984a393a92f8a)
Kets
For any given vector
the notation
is used to denote
; that is,
![{\displaystyle \mid g\rangle :=g.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b48d0cc63a3079261f4fbc0b3283d1279dd2999)
The assignment
is just the identity map
which is why
holds for all
and all scalars
The notation
and
is used in place of
and
respectively. As expected,
and
really is just the scalar
Adjoints and transposes
Let
be a continuous linear operator between Hilbert spaces
and
As before, let
and
Denote by
![{\displaystyle {\begin{alignedat}{4}\Phi _{H}:\;&&H&&\;\to \;&H^{*}\\[0.3ex]&&g&&\;\mapsto \;&\langle \,g\mid \cdot \,\rangle _{H}\\\end{alignedat}}\quad {\text{ and }}\quad {\begin{alignedat}{4}\Phi _{Z}:\;&&Z&&\;\to \;&Z^{*}\\[0.3ex]&&y&&\;\mapsto \;&\langle \,y\mid \cdot \,\rangle _{Z}\\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/779c55ea574305ef9fcab3fef4790eb65bb7451c)
the usual bijective antilinear isometries that satisfy:
![{\displaystyle \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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eef73fc6b1b31ec43561660f06ca47a4e92ecb96)
Definition of the adjoint
For every
the scalar-valued map
[note 7] on
defined by
![{\displaystyle h\mapsto \langle z\mid Ah\rangle _{Z}=\langle Ah,z\rangle _{Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69f13df7fea15b07f92d743e0cd23c47cdb45826)
is a continuous linear functional on
and so by the Riesz representation theorem, there exists a unique vector in
denoted by
such that
or equivalently, such that
![{\displaystyle \langle z\mid Ah\rangle _{Z}=\left\langle A^{*}z\mid h\right\rangle _{H}\quad {\text{ for all }}h\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9cacae8eac4b38d8824fb1c418dca48fce1ad5f)
The assignment
thus induces a function
called the adjoint of
whose defining condition is
![{\displaystyle \langle z\mid Ah\rangle _{Z}=\left\langle A^{*}z\mid h\right\rangle _{H}\quad {\text{ for all }}h\in H{\text{ and all }}z\in Z.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee11c42b3239073ae017939102edb292357204f7)
The adjoint
![{\displaystyle A^{*}:Z\to H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e4cba4209e55fa2f39d001316e622e627bbd8d4)
is necessarily a
continuous (equivalently, a
bounded)
linear operator.
If
is finite dimensional with the standard inner product and if
is the transformation matrix of
with respect to the standard orthonormal basis then
's conjugate transpose
is the transformation matrix of the adjoint
Adjoints are transposes
It is also possible to define the transpose or algebraic adjoint of
which is the map
defined by sending a continuous linear functionals
to
![{\displaystyle {}^{t}A(\psi ):=\psi \circ A,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7cb040291e99e7d2b6c8edf74bbb31bd4be4fa)
where the
composition ![{\displaystyle \psi \circ A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f12225f4492b5bc2460ccb79f84c12ef41d7efa1)
is always a continuous linear functional on
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
and it satisfies
![{\displaystyle \|A\|=\left\|{}^{t}A\right\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fbfb455ed74e4e1b4ab074d6f29b54f4b6c0065)
(this is true more generally, when
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
and
![{\displaystyle Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd)
are merely
normed spaces). So for example, if
![{\displaystyle z\in Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3472db536f24864ab268d42ae277fcfc90d2998)
then
![{\displaystyle {}^{t}A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/038c833f94b6e71efbdbfc30694ea1dcf7b8f523)
sends the continuous linear functional
![{\displaystyle \langle z\mid \cdot \rangle _{Z}\in Z^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21ae0a60f2f0809aeb047a6a249ab1cf1c407e14)
(defined on
![{\displaystyle Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd)
by
![{\displaystyle g\mapsto \langle z\mid g\rangle _{Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46d99e2b72f2d7bc4df25437280e73ce31e3a043)
) to the continuous linear functional
![{\displaystyle \langle z\mid A(\cdot )\rangle _{Z}\in H^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd474e387df6f5961e5d9401cb2202fd4107885d)
(defined on
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
by
![{\displaystyle h\mapsto \langle z\mid A(h)\rangle _{Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37a5854232a31716f1c405c830bf8994ac429b84)
);
[note 7] using bra-ket notation, this can be written as
![{\displaystyle {}^{t}A\langle z\mid ~=~\langle z\mid A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/391ec0fcc045654336e96a904d7a804f5932d37b)
where the juxtaposition of
![{\displaystyle \langle z\mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf0820cfc2e7bcde95601fc8cb03770583f593c7)
with
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
on the right hand side denotes function composition:
The adjoint
is actually just to the transpose
when the Riesz representation theorem is used to identify
with
and
with
Explicitly, the relationship between the adjoint and transpose is:
![{\displaystyle {}^{t}A~\circ ~\Phi _{Z}~=~\Phi _{H}~\circ ~A^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa24b38e3a3abbded03e4fe4d7c06b911713bf8b) | | (Adjoint-transpose) |
which can be rewritten as:
![{\displaystyle A^{*}~=~\Phi _{H}^{-1}~\circ ~{}^{t}A~\circ ~\Phi _{Z}\quad {\text{ and }}\quad {}^{t}A~=~\Phi _{H}~\circ ~A^{*}~\circ ~\Phi _{Z}^{-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46a0594e1f0e6b15fbcd4cc75721b9e9a302178b)
Alternatively, the value of the left and right hand sides of (Adjoint-transpose) at any given
can be rewritten in terms of the inner products as:
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be694b5977a8614b85352eec9be25db7b3d4e2f9)
so that
![{\displaystyle {}^{t}A~\circ ~\Phi _{Z}~=~\Phi _{H}~\circ ~A^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa24b38e3a3abbded03e4fe4d7c06b911713bf8b)
holds if and only if
![{\displaystyle \langle z\mid A(\cdot )\rangle _{Z}=\langle A^{*}z\mid \cdot \,\rangle _{H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9542ce3378829b8aa3b466c692434b5556089edf)
holds; but the equality on the right holds by definition of
![{\displaystyle A^{*}z.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26535d8749a07c40fc1aef96dbc71b6559f9d8d0)
The defining condition of
![{\displaystyle A^{*}z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8728c8b918271d35a6ce243fb2b8ccafac268117)
can also be written
![{\displaystyle \langle z\mid A~=~\langle A^{*}z\mid }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb9b81e993cb7a216d8bd6a4f3d5bf03ab810468)
if bra-ket notation is used.
Descriptions of self-adjoint, normal, and unitary operators
Assume
and let
Let
be a continuous (that is, bounded) linear operator.
Whether or not
is self-adjoint, normal, or unitary depends entirely on whether or not
satisfies certain defining conditions related to its adjoint, which was shown by (Adjoint-transpose) to essentially be just the transpose
Because the transpose of
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
that can be defined entirely in terms of
the inner product
on
and some given vector
Specifically, these are
and
[note 7] where
![{\displaystyle \left\langle Ah\mid \cdot \,\right\rangle =\Phi (Ah)=(\Phi \circ A)h\quad {\text{ and }}\quad \langle h\mid A(\cdot )\rangle =\left({}^{t}A\circ \Phi \right)h.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/969d0e54fe2e84ecf71c75b751da48fd2baa4722)
Self-adjoint operators
A continuous linear operator
is called self-adjoint it is equal to its own adjoint; that is, if
Using (Adjoint-transpose), this happens if and only if:
![{\displaystyle \Phi \circ A={}^{t}A\circ \Phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c07c61dad61857bcf64474ef4405a64284232f94)
where this equality can be rewritten in the following two equivalent forms:
![{\displaystyle A=\Phi ^{-1}\circ {}^{t}A\circ \Phi \quad {\text{ or }}\quad {}^{t}A=\Phi \circ A\circ \Phi ^{-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66c6ec0e7759898787e4c47b41573f71ae5a2001)
Unraveling notation and definitions produces the following characterization of self-adjoint operators in terms of the aforementioned continuous linear functionals:
is self-adjoint if and only if for all
the linear functional
[note 7] is equal to the linear functional
; that is, if and only if
![{\displaystyle \langle z\mid A(\cdot )\rangle =\langle Az\mid \cdot \,\rangle \quad {\text{ for all }}z\in H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58f4df6ffee3eecb0cd83fba8dd96046e84b8330) | | (Self-adjointness functionals) |
where if bra-ket notation is used, this is
![{\displaystyle \langle z\mid A~=~\langle Az\mid \quad {\text{ for all }}z\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/326229db8048675a3813dfab587cf48dff8a52c1)
Normal operators
A continuous linear operator
is called normal if
which happens if and only if for all
![{\displaystyle \left\langle AA^{*}z\mid h\right\rangle =\left\langle A^{*}Az\mid h\right\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a0e5dc6fb7bc45a9a1def155d3848821e721ead)
Using (Adjoint-transpose) and unraveling notation and definitions produces[proof 2] the following characterization of normal operators in terms of inner products of continuous linear functionals:
is a normal operator if and only if
![{\displaystyle \left\langle \,\langle Ah\mid \cdot \,\rangle \mid \langle Az\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b1d0026d6ac74b5cad700acf2196a591a80d50) | | (Normality functionals) |
where the left hand side is also equal to
The left hand side of this characterization involves only linear functionals of the form
while the right hand side involves only linear functions of the form
(defined as above[note 7]). So in plain English, characterization (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
for both forms). In other words, if it happens to be the case (and when
is injective or self-adjoint, it is) that the assignment of linear functionals
is well-defined (or alternatively, if
is well-defined) where
ranges over
then
is a normal operator if and only if this assignment preserves the inner product on
The fact that every self-adjoint bounded linear operator is normal follows readily by direct substitution of
into either side of
This same fact also follows immediately from the direct substitution of the equalities (Self-adjointness functionals) into either side of (Normality functionals).
Alternatively, for a complex Hilbert space, the continuous linear operator
is a normal operator if and only if
for every
which happens if and only if
![{\displaystyle \|Az\|_{H}=\|\langle z\,|\,A(\cdot )\rangle \|_{H^{*}}\quad {\text{ for every }}z\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3533bdad631e9abca3b7a4a73c70647e9b29548f)
Unitary operators
An invertible bounded linear operator
is said to be unitary if its inverse is its adjoint:
By using (Adjoint-transpose), this is seen to be equivalent to
Unraveling notation and definitions, it follows that
is unitary if and only if
![{\displaystyle \langle A^{-1}z\mid \cdot \,\rangle =\langle z\mid A(\cdot )\rangle \quad {\text{ for all }}z\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77ddd7e2a4de9311b1ad3d2b52439175e05c8eff)
The fact that a bounded invertible linear operator
is unitary if and only if
(or equivalently,
) produces another (well-known) characterization: an invertible bounded linear map
is unitary if and only if
![{\displaystyle \langle Az\mid A(\cdot )\,\rangle =\langle z\mid \cdot \,\rangle \quad {\text{ for all }}z\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dba9a10175f191e1a766d46a30c1aaae1e7588c)
Because
is invertible (and so in particular a bijection), this is also true of the transpose
This fact also allows the vector
in the above characterizations to be replaced with
or
thereby producing many more equalities. Similarly,
can be replaced with
or
See also
Citations
Notes
- ^ If
then the inner product will be symmetric so it does not matter which coordinate of the inner product the element
is placed into because the same map will result. But if
then except for the constant
map, antilinear functionals on
are completely distinct from linear functionals on
which makes the coordinate that
is placed into is very important. For a non-zero
to induce a linear functional (rather than an antilinear functional),
must be placed into the antilinear 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
which is not a linear functional on
and so it will not be an element of the continuous dual space
- ^ This means that for all vectors
(1)
is injective. (2) The norms of
and
are the same:
(3)
is an additive map, meaning that
for all
(4)
is conjugate homogeneous:
for all scalars
(5)
is real homogeneous:
for all real numbers
- ^ a b This footnote explains how to define - using only
's operations - addition and scalar multiplication of affine hyperplanes so that these operations correspond to addition and scalar multiplication of linear functionals. Let
be any vector space and let
denote its algebraic dual space. Let
and let
and
denote the (unique) vector space operations on
that make the bijection
defined by
into a vector space isomorphism. Note that
if and only if
so
is the additive identity of
(because this is true of
in
and
is a vector space isomorphism). For every
let
if
and let
otherwise; if
then
so this definition is consistent with the usual definition of the kernel of a linear functional. Say that
are parallel if
where if
and
are not empty then this happens if and only if the linear functionals
and
are non-zero scalar multiples of each other. The vector space operations on the vector space of affine hyperplanes
are now described in a way that involves only the vector space operations on
; this results in an interpretation of the vector space operations on the algebraic dual space
that is entirely in terms of affine hyperplanes. Fix hyperplanes
If
is a scalar then
Describing the operation
in terms of only the sets
and
is more complicated because by definition,
If
(respectively, if
) then
is equal to
(resp. is equal to
) so assume
and
The hyperplanes
and
are parallel if and only if there exists some scalar
(necessarily non-0) such that
in which case
this can optionally be subdivided into two cases: if
(which happens if and only if the linear functionals
and
are negatives of each) then
while if
then
Finally, assume now that
Then
is the unique affine hyperplane containing both
and
as subsets; explicitly,
and
To see why this formula for
should hold, consider
and
where
and
(or alternatively,
). Then by definition,
and
Now
is an affine subspace of codimension
in
(it is equal to a translation of the
-axis
). The same is true of
Plotting an
-
-plane cross section (that is, setting
constant) of the sets
and
(each of which will be plotted as a line), the set
will then be plotted as the (unique) line passing through the
and
(which will be plotted as two distinct points) while
will be plotted the line through the origin that is parallel to
The above formulas for
and
follow naturally from the plot and they also hold in general. - ^ Showing that there is a non-zero vector
in
relies on the continuity of
and the Cauchy completeness of
This is the only place in the proof in which these properties are used. - ^ Technically,
means that the addition map
defined by
is a surjective linear isomorphism and homeomorphism. See the article on complemented subspaces for more details. - ^ The usual notation for plugging an element
into a linear map
is
and sometimes
Replacing
with
produces
or
which is unsightly (despite being consistent with the usual notation used with functions). Consequently, the symbol
is appended to the end, so that the notation
is used instead to denote this value
- ^ a b c d e The notation
denotes the continuous linear functional defined by
Proofs
- ^ This is because
Now use
and
and solve for
- ^
where
and
By definition of the adjoint,
so taking the complex conjugate of both sides proves that
From
it follows that
where
and
Bibliography
- Bachman, George; Narici, Lawrence (2000). Functional Analysis (Second ed.). Mineola, New York: Dover Publications. ISBN 978-0486402512. OCLC 829157984.
- Fréchet, M. (1907). "Sur les ensembles de fonctions et les opérations linéaires". Les Comptes rendus de l'Académie des sciences (in French). 144: 1414–1416.
- 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).
- Riesz, F. (1907). "Sur une espèce de géométrie analytique des systèmes de fonctions sommables". Comptes rendus de l'Académie des Sciences (in French). 144: 1409–1411.
- Riesz, F. (1909). "Sur les opérations fonctionnelles linéaires". Comptes rendus de l'Académie des Sciences (in French). 149: 974–977.
- Roman, Stephen (2008), Advanced Linear Algebra, Graduate Texts in Mathematics (Third ed.), Springer, ISBN 978-0-387-72828-5
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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