Riesz representation theorem

Riesz representation theorem This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem. For more theorems that are sometimes called Riesz's theorem, see Riesz theorem.

The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field 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.

Contents 1 Preliminaries and notation 1.1 Linear and antilinear maps 1.2 Mathematics vs. physics notations and definitions of inner product 1.3 Canonical norm and inner product on the dual space and anti-dual space 2 Riesz representation theorem 2.1 Statement 2.2 Observations 2.3 Constructions of the representing vector 2.3.1 Example in finite dimensions using matrix transformations 2.4 Relationship with the associated real Hilbert space 2.5 Canonical injections into the dual and anti-dual 3 Extending the bra–ket notation to bras and kets 4 Adjoints and transposes 4.1 Definition of the adjoint 4.2 Adjoints are transposes 4.3 Descriptions of self-adjoint, normal, and unitary operators 5 See also 6 Citations 7 Notes 8 Bibliography Preliminaries and notation Let {displaystyle H} be a Hilbert space over a field {displaystyle mathbb {F} ,} where {displaystyle mathbb {F} } is either the real numbers {displaystyle mathbb {R} } or the complex numbers {displaystyle mathbb {C} .} If {displaystyle mathbb {F} =mathbb {C} } (resp. if {displaystyle mathbb {F} =mathbb {R} } ) then {displaystyle 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 {displaystyle 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 By definition, an antilinear map (also called a conjugate-linear map) {displaystyle f:Hto Y} is a map between vector spaces that is additive: {displaystyle f(x+y)=f(x)+f(y)quad {text{ for all }}x,yin H,} and antilinear (also called conjugate-linear or conjugate-homogeneous): {displaystyle f(cx)={overline {c}}f(x)quad {text{ for all }}xin H{text{ and all scalar }}cin mathbb {F} .} In contrast, a map {displaystyle f:Hto Y} is linear if it is additive and homogeneous: {displaystyle f(cx)=cf(x)quad {text{ for all }}xin Hquad {text{ and all scalars }}cin mathbb {F} .} Every constant {displaystyle 0} map is always both linear and antilinear. If {displaystyle 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 {displaystyle H} is a function {displaystyle Hto mathbb {F} } whose codomain is the underlying scalar field {displaystyle mathbb {F} .} Denote by {displaystyle H^{*}} (resp. by {displaystyle {overline {H}}^{*})} the set of all continuous linear (resp. continuous antilinear) functionals on {displaystyle H,} which is called the (continuous) dual space (resp. the (continuous) anti-dual space) of {displaystyle H.} [1] If {displaystyle mathbb {F} =mathbb {R} } then linear functionals on {displaystyle H} are the same as antilinear functionals and consequently, the same is true for such continuous maps: that is, {displaystyle H^{*}={overline {H}}^{*}.} One-to-one correspondence between linear and antilinear functionals Given any functional {displaystyle f~:~Hto mathbb {F} ,} the conjugate of {displaystyle f} is the functional {displaystyle {begin{alignedat}{4}{overline {f}}:,&H&&to ,&&mathbb {F} \&h&&mapsto ,&&{overline {f(h)}}.\end{alignedat}}} This assignment is most useful when {displaystyle mathbb {F} =mathbb {C} } because if {displaystyle mathbb {F} =mathbb {R} } then {displaystyle f={overline {f}}} and the assignment {displaystyle fmapsto {overline {f}}} reduces down to the identity map.

The assignment {displaystyle fmapsto {overline {f}}} defines an antilinear bijective correspondence from the set of all functionals (resp. all linear functionals, all continuous linear functionals {displaystyle H^{*}} ) on {displaystyle H,} onto the set of all functionals (resp. all antilinear functionals, all continuous antilinear functionals {displaystyle {overline {H}}^{*}} ) on {displaystyle H.} Mathematics vs. physics notations and definitions of inner product The Hilbert space {displaystyle H} has an associated inner product {displaystyle Htimes Hto mathbb {F} } valued in {displaystyle H} 's underlying scalar field {displaystyle mathbb {F} } that is linear in one coordinate and antilinear in the other (as described in detail below). If {displaystyle H} is a complex Hilbert space (meaning, if {displaystyle 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 {displaystyle 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 {displaystyle H} is often denoted by {displaystyle leftlangle cdot ,cdot rightrangle } or {displaystyle leftlangle cdot ,cdot rightrangle _{H}} while in physics, the bra–ket notation {displaystyle leftlangle cdot mid cdot rightrangle } or {displaystyle leftlangle cdot mid cdot rightrangle _{H}} is typically used instead. In this article, these two notations will be related by the equality: {displaystyle leftlangle x,yrightrangle :=leftlangle ymid xrightrangle quad {text{ for all }}x,yin H.} Competing definitions of the inner product The maps {displaystyle leftlangle cdot ,cdot rightrangle } and {displaystyle leftlangle cdot mid cdot rightrangle } are assumed to have the following two properties: The map {displaystyle leftlangle cdot ,cdot rightrangle } is linear in its first coordinate; equivalently, the map {displaystyle leftlangle cdot mid cdot rightrangle } is linear in its second coordinate. Explicitly, this means that for every fixed {displaystyle yin H,} the map that is denoted by {displaystyle leftlangle ,ymid cdot ,rightrangle =leftlangle ,cdot ,y,rightrangle :Hto mathbb {F} } and defined by {displaystyle hmapsto leftlangle ,ymid h,rightrangle =leftlangle ,h,y,rightrangle quad {text{ for all }}hin H} is a linear functional on {displaystyle H.} In fact, this linear functional is continuous, so {displaystyle leftlangle ,ymid cdot ,rightrangle =leftlangle ,cdot ,y,rightrangle in H^{*}.} The map {displaystyle leftlangle cdot ,cdot rightrangle } is antilinear in its second coordinate; equivalently, the map {displaystyle leftlangle cdot mid cdot rightrangle } is antilinear in its first coordinate. Explicitly, this means that for every fixed {displaystyle yin H,} the map that is denoted by {displaystyle leftlangle ,cdot mid y,rightrangle =leftlangle ,y,cdot ,rightrangle :Hto mathbb {F} } and defined by {displaystyle hmapsto leftlangle ,hmid y,rightrangle =leftlangle ,y,h,rightrangle quad {text{ for all }}hin H} is an antilinear functional on {displaystyle H.} In fact, this antilinear functional is continuous, so {displaystyle leftlangle ,cdot mid y,rightrangle =leftlangle ,y,cdot ,rightrangle 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 {displaystyle leftlangle cdot ,cdot rightrangle } 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 {displaystyle leftlangle cdot |cdot rightrangle } 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 {displaystyle x=y} then {displaystyle langle ,xmid x,rangle =langle ,x,x,rangle } is a non-negative real number and the map {displaystyle |x|:={sqrt {langle x,xrangle }}={sqrt {langle xmid xrangle }}} defines a canonical norm on {displaystyle H} that makes {displaystyle H} into a normed space.[1] As with all normed spaces, the (continuous) dual space {displaystyle H^{*}} carries a canonical norm, called the dual norm, that is defined by[1] {displaystyle |f|_{H^{*}}~:=~sup _{|x|leq 1,xin H}|f(x)|quad {text{ for every }}fin H^{*}.} The canonical norm on the (continuous) anti-dual space {displaystyle {overline {H}}^{*},} denoted by {displaystyle |f|_{{overline {H}}^{*}},} is defined by using this same equation:[1] {displaystyle |f|_{{overline {H}}^{*}}~:=~sup _{|x|leq 1,xin H}|f(x)|quad {text{ for every }}fin {overline {H}}^{*}.} This canonical norm on {displaystyle H^{*}} satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on {displaystyle H^{*},} which this article will denote by the notations {displaystyle leftlangle f,grightrangle _{H^{*}}:=leftlangle gmid frightrangle _{H^{*}},} where this inner product turns {displaystyle H^{*}} into a Hilbert space. There are now two ways of defining a norm on {displaystyle H^{*}:} the norm induced by this inner product (that is, the norm defined by {displaystyle fmapsto {sqrt {leftlangle f,frightrangle _{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 {displaystyle fin H^{*}:} {displaystyle sup _{|x|leq 1,xin H}|f(x)|=|f|_{H^{*}}~=~{sqrt {langle f,frangle _{H^{*}}}}~=~{sqrt {langle fmid frangle _{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 {displaystyle H^{*}.} The same equations that were used above can also be used to define a norm and inner product on {displaystyle H} 's anti-dual space {displaystyle {overline {H}}^{*}.} [1] Canonical isometry between the dual and antidual The complex conjugate {displaystyle {overline {f}}} of a functional {displaystyle f,} 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}}^{*}}} for every {displaystyle fin H^{*}} and every {displaystyle gin {overline {H}}^{*}.} 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}}} as well as its inverse {displaystyle operatorname {Cong} ^{-1}~:~{overline {H}}^{*}to H^{*}} are antilinear isometries and consequently also homeomorphisms. The inner products on the dual space {displaystyle H^{*}} and the anti-dual space {displaystyle {overline {H}}^{*},} denoted respectively by {displaystyle langle ,cdot ,,,cdot ,rangle _{H^{*}}} and {displaystyle langle ,cdot ,,,cdot ,rangle _{{overline {H}}^{*}},} 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,gin H^{*}} 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,gin {overline {H}}^{*}.} If {displaystyle mathbb {F} =mathbb {R} } then {displaystyle H^{*}={overline {H}}^{*}} and this canonical map {displaystyle operatorname {Cong} :H^{*}to {overline {H}}^{*}} reduces down to the identity map.

Riesz representation theorem Two vectors {displaystyle x} and {displaystyle y} are orthogonal if {displaystyle langle x,yrangle =0,} which happens if and only if {displaystyle |y|leq |y+sx|} for all scalars {displaystyle s.} [2] The orthogonal complement of a subset {displaystyle Csubseteq H} is {displaystyle C^{bot }:={,yin H:langle y,crangle =0{text{ for all }}cin C,},} which is always a closed vector subspace of {displaystyle H.} The Hilbert projection theorem guarantees that for any nonempty closed convex subset {displaystyle C} of a Hilbert space there exists a unique vector {displaystyle min C} such that {displaystyle |m|=inf _{cin C}|c|;} that is, {displaystyle min C} is the (unique) global minimum point of the function {displaystyle Cto [0,infty )} defined by {displaystyle cmapsto |c|.} Statement Theorem: Let {displaystyle H} be a Hilbert space whose inner product {displaystyle leftlangle x,yrightrangle } is linear in its first argument and antilinear in its second argument (cf the physics notation {displaystyle langle ymid xrangle :=langle x,yrangle } ). For every continuous linear functional {displaystyle varphi in H^{*},} there exists a unique {displaystyle f_{varphi }in H} , called the Riesz representation of {displaystyle varphi ,} such that {displaystyle varphi (x)=leftlangle x,f_{varphi }rightrangle =leftlangle f_{varphi }mid xrightrangle quad {text{ for all }}xin H.} Note that for complex Hilbert spaces, {displaystyle f_{varphi }} is always located in the antilinear coordinate of the inner product.[note 1] Moreover, the length of the representation vector is equal to the norm of the functional: {displaystyle left|f_{varphi }right|_{H}=|varphi |_{H^{*}},} and {displaystyle f_{varphi }} is the unique vector {displaystyle f_{varphi }in left(ker varphi right)^{bot }} with {displaystyle varphi left(f_{varphi }right)=|varphi |^{2}.} Any non-zero {displaystyle qin (ker varphi )^{bot }} can be written as {displaystyle q=left(|q|^{2}/,{overline {varphi (q)}}right) f_{varphi }.} Furthermore, with regard to the Hilbert projection theorem, {displaystyle f_{varphi }} is the unique element of minimum norm in {displaystyle C:=varphi ^{-1}left(|varphi |^{2}right)} ; i.e., the unique element of {displaystyle C} with {displaystyle left|f_{varphi }right|=inf _{cin C}|c|.} Corollary — The canonical map from {displaystyle H} into its dual {displaystyle H^{*}} [1] is the injective antilinear operator isometry[note 2][1] {displaystyle {begin{alignedat}{4}Phi :;&&H&&;to ;&H^{*}\[0.3ex]&&y&&;mapsto ;&langle ,cdot ,,yrangle =langle y|,cdot ,rangle \end{alignedat}}} The Riesz representation theorem states that this map is surjective (and thus bijective) when {displaystyle H} 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}}.} Consequently, every continuous linear functional on the Hilbert space {displaystyle H} can be written uniquely in the form {displaystyle langle y,|,cdot ,rangle } [1] where {displaystyle |langle y,|cdot rangle |_{H^{*}}=|y|_{H}} for every {displaystyle yin H.} The assignment {displaystyle ymapsto langle y,cdot rangle =langle cdot ,|,yrangle } can also be viewed as a bijective linear isometry {displaystyle Hto {overline {H}}^{*}} into the anti-dual space of {displaystyle H,} [1] which is the complex conjugate vector space of the continuous dual space {displaystyle H^{*}.} The inner products on {displaystyle H} and {displaystyle H^{*}} are related by {displaystyle leftlangle Phi h,Phi krightrangle _{H^{*}}={overline {langle h,krangle }}_{H}=langle k,hrangle _{H}quad {text{ for all }}h,kin H} and similarly, {displaystyle leftlangle Phi ^{-1}varphi ,Phi ^{-1}psi rightrangle _{H}={overline {langle varphi ,psi rangle }}_{H^{*}}=leftlangle psi ,varphi rightrangle _{H^{*}}quad {text{ for all }}varphi ,psi in H^{*}.} The set {displaystyle C:=varphi ^{-1}left(|varphi |^{2}right)} satisfies {displaystyle C=f_{varphi }+ker varphi } and {displaystyle C-C=ker varphi } so when {displaystyle f_{varphi }neq 0} then {displaystyle C} can be interpreted as being the affine hyperplane[note 3] that is parallel to the vector subspace {displaystyle ker varphi } and contains {displaystyle f_{varphi }.} For {displaystyle yin H,} the physics notation for the functional {displaystyle Phi (y)in H^{*}} is the bra {displaystyle langle y|,} where explicitly this means that {displaystyle langle y|:=Phi (y),} which complements the ket notation {displaystyle |yrangle } defined by {displaystyle |yrangle :=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 {displaystyle langle psi ,|} has a corresponding ket {displaystyle |,psi rangle ,} and the latter is unique.

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

show Proof[3] Observations If {displaystyle varphi in H^{*}} then {displaystyle varphi left(f_{varphi }right)=leftlangle f_{varphi },f_{varphi }rightrangle =left|f_{varphi }right|^{2}=|varphi |^{2}.} So in particular, {displaystyle varphi left(f_{varphi }right)geq 0} is always real and furthermore, {displaystyle varphi left(f_{varphi }right)=0} if and only if {displaystyle f_{varphi }=0} if and only if {displaystyle varphi =0.} Linear functionals as affine hyperplanes A non-trivial continuous linear functional {displaystyle varphi } is often interpreted geometrically by identifying it with the affine hyperplane {displaystyle A:=varphi ^{-1}(1)} (the kernel {displaystyle ker varphi =varphi ^{-1}(0)} is also often visualized alongside {displaystyle A:=varphi ^{-1}(1)} although knowing {displaystyle A} is enough to reconstruct {displaystyle ker varphi } because if {displaystyle A=varnothing } then {displaystyle ker varphi =H} and otherwise {displaystyle ker varphi =A-A} ). In particular, the norm of {displaystyle varphi } should somehow be interpretable as the "norm of the hyperplane {displaystyle A} ". When {displaystyle varphi neq 0} then the Riesz representation theorem provides such an interpretation of {displaystyle |varphi |} in terms of the affine hyperplane[note 3] {displaystyle A:=varphi ^{-1}(1)} as follows: using the notation from the theorem's statement, from {displaystyle |varphi |^{2}neq 0} it follows that {displaystyle C:=varphi ^{-1}left(|varphi |^{2}right)=|varphi |^{2}varphi ^{-1}(1)=|varphi |^{2}A} and so {displaystyle |varphi |=left|f_{varphi }right|=inf _{cin C}|c|} implies {displaystyle |varphi |=inf _{ain A}|varphi |^{2}|a|} and thus {displaystyle |varphi |={frac {1}{inf _{ain A}|a|}}.} This can also be seen by applying the Hilbert projection theorem to {displaystyle A} and concluding that the global minimum point of the map {displaystyle Ato [0,infty )} defined by {displaystyle amapsto |a|} is {displaystyle {frac {f_{varphi }}{|varphi |^{2}}}in A.} The formulas {displaystyle {frac {1}{inf _{ain A}|a|}}=sup _{ain A}{frac {1}{|a|}}} provide the promised interpretation of the linear functional's norm {displaystyle |varphi |} entirely in terms of its associated affine hyperplane {displaystyle A=varphi ^{-1}(1)} (because with this formula, knowing only the set {displaystyle A} is enough to describe the norm of its associated linear functional). Defining {displaystyle {frac {1}{infty }}:=0,} the infimum formula {displaystyle |varphi |={frac {1}{inf _{ain varphi ^{-1}(1)}|a|}}} will also hold when {displaystyle varphi =0.} When the supremum is taken in {displaystyle mathbb {R} } (as is typically assumed), then the supremum of the empty set is {displaystyle sup varnothing =-infty } but if the supremum is taken the non-negative reals {displaystyle [0,infty )} (which is the image/range of the norm {displaystyle |,cdot ,|} when {displaystyle dim H>0} ) then this supremum is instead {displaystyle sup varnothing =0,} in which case the supremum formula {displaystyle |varphi |=sup _{ain varphi ^{-1}(1)}{frac {1}{|a|}}} will also hold when {displaystyle varphi =0} (although the atypical equality {displaystyle sup varnothing =0} is usually unexpected and so risks causing confusion).

Constructions of the representing vector Using the notation from the theorem above, several ways of constructing {displaystyle f_{varphi }} from {displaystyle varphi in H^{*}} are now described. If {displaystyle varphi =0} then {displaystyle f_{varphi }:=0} ; in other words, {displaystyle f_{0}=0.} This special case of {displaystyle varphi =0} is henceforth assumed to be known, which is why some of the constructions given below start by assuming {displaystyle varphi neq 0.} Orthogonal complement of kernel If {displaystyle varphi neq 0} then for any {displaystyle 0neq uin (ker varphi )^{bot },} {displaystyle f_{varphi }:={frac {{overline {varphi (u)}}u}{|u|^{2}}}.} If {displaystyle uin (ker varphi )^{bot }} is a unit vector (meaning {displaystyle |u|=1} ) then {displaystyle f_{varphi }:={overline {varphi (u)}}u} (this is true even if {displaystyle varphi =0} because in this case {displaystyle f_{varphi }={overline {varphi (u)}}u={overline {0}}u=0} ). If {displaystyle u} is a unit vector satisfying the above condition then the same is true of {displaystyle -u,} which is also a unit vector in {displaystyle (ker varphi )^{bot }.} However, {displaystyle {overline {varphi (-u)}}(-u)={overline {varphi (u)}}u=f_{varphi }} so both these vectors result in the same {displaystyle f_{varphi }.} Orthogonal projection onto kernel If {displaystyle xin H} is such that {displaystyle varphi (x)neq 0} and if {displaystyle x_{K}} is the orthogonal projection of {displaystyle x} onto {displaystyle ker varphi } then[proof 1] {displaystyle f_{varphi }={frac {|varphi |^{2}}{varphi (x)}}left(x-x_{K}right).} Orthonormal basis Given an orthonormal basis {displaystyle left{e_{i}right}_{iin I}} of {displaystyle H} and a continuous linear functional {displaystyle varphi in H^{*},} the vector {displaystyle f_{varphi }in H} can be constructed uniquely by {displaystyle f_{varphi }=sum _{iin I}{overline {varphi left(e_{i}right)}}e_{i}} where all but at most countably many {displaystyle varphi left(e_{i}right)} will be equal to {displaystyle 0} and where the value of {displaystyle f_{varphi }} does not actually depend on choice of orthonormal basis (that is, using any other orthonormal basis for {displaystyle H} will result in the same vector). If {displaystyle yin H} is written as {displaystyle y=sum _{iin I}a_{i}e_{i}} then {displaystyle varphi (y)=sum _{iin I}varphi left(e_{i}right)a_{i}=langle f_{varphi }|yrangle } and {displaystyle left|f_{varphi }right|^{2}=varphi left(f_{varphi }right)=sum _{iin I}varphi left(e_{i}right){overline {varphi left(e_{i}right)}}=sum _{iin I}left|varphi left(e_{i}right)right|^{2}=|varphi |^{2}.} If the orthonormal basis {displaystyle left{e_{i}right}_{iin I}=left{e_{i}right}_{i=1}^{infty }} 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 } and if {displaystyle yin H} is written as {displaystyle y=sum _{iin I}a_{i}e_{i}=a_{1}e_{1}+a_{2}e_{2}+cdots } then {displaystyle varphi (y)=varphi left(e_{1}right)a_{1}+varphi left(e_{2}right)a_{2}+cdots =langle f_{varphi }|yrangle .} Example in finite dimensions using matrix transformations Consider the special case of {displaystyle H=mathbb {C} ^{n}} (where {displaystyle n>0} is an integer) with the standard inner product {displaystyle langle zmid wrangle :={overline {,{vec {z}},,}}^{operatorname {T} }{vec {w}}qquad {text{ for all }};w,zin H} where {displaystyle w{text{ and }}z} are represented as column matrices {displaystyle {vec {w}}:={begin{bmatrix}w_{1}\vdots \w_{n}end{bmatrix}}} and {displaystyle {vec {z}}:={begin{bmatrix}z_{1}\vdots \z_{n}end{bmatrix}}} with respect to the standard orthonormal basis {displaystyle e_{1},ldots ,e_{n}} on {displaystyle H} (here, {displaystyle e_{i}} is {displaystyle 1} at its {displaystyle i} th coordinate and {displaystyle 0} everywhere else; as usual, {displaystyle H^{*}} 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]} denotes the conjugate transpose of {displaystyle {vec {z}}.} Let {displaystyle varphi in H^{*}} be any linear functional and let {displaystyle varphi _{1},ldots ,varphi _{n}in mathbb {C} } 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,} where it can be shown that {displaystyle varphi _{i}=varphi left(e_{i}right)} for all {displaystyle i=1,ldots ,n.} Then the Riesz representation of {displaystyle varphi } 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.} To see why, identify every vector {displaystyle w=left(w_{1},ldots ,w_{n}right)} in {displaystyle H} with the column matrix {displaystyle {vec {w}}:={begin{bmatrix}w_{1}\vdots \w_{n}end{bmatrix}}} so that {displaystyle f_{varphi }} 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}}.} As usual, also identify the linear functional {displaystyle varphi } with its transformation matrix, which is the row matrix {displaystyle {vec {varphi }}:=left[varphi _{1},ldots ,varphi _{n}right]} so that {displaystyle {vec {f_{varphi }}}:={overline {,{vec {varphi }},,}}^{operatorname {T} }} and the function {displaystyle varphi } is the assignment {displaystyle {vec {w}}mapsto {vec {varphi }},{vec {w}},} where the right hand side is matrix multiplication. Then for all {displaystyle w=left(w_{1},ldots ,w_{n}right)in H,} {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}}=leftlangle ,,f_{varphi },mid ,w,rightrangle ,} which shows that {displaystyle f_{varphi }} satisfies the defining condition of the Riesz representation of {displaystyle varphi .} The bijective antilinear isometry {displaystyle Phi :Hto H^{*}} 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} to the linear functional {displaystyle Phi (z)in H^{*}} on {displaystyle H} 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},} where under the identification of vectors in {displaystyle H} with column matrices and vector in {displaystyle H^{*}} with row matrices, {displaystyle Phi } 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].} As described in the corollary, {displaystyle Phi } 's inverse {displaystyle Phi ^{-1}:H^{*}to H} is the antilinear isometry {displaystyle varphi mapsto f_{varphi },} 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);} where in terms of matrices, {displaystyle Phi ^{-1}} 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}}.} Thus in terms of matrices, each of {displaystyle Phi :Hto H^{*}} and {displaystyle Phi ^{-1}:H^{*}to H} is just the operation of conjugate transposition {displaystyle {vec {v}}mapsto {overline {,{vec {v}},}}^{operatorname {T} }} (although between different spaces of matrices: if {displaystyle H} is identified with the space of all column (respectively, row) matrices then {displaystyle H^{*}} is identified with the space of all row (respectively, column) matrices).

This example used the standard inner product, which is the map {displaystyle langle zmid wrangle :={overline {,{vec {z}},,}}^{operatorname {T} }{vec {w}},} but if a different inner product is used, such as {displaystyle langle zmid wrangle _{M}:={overline {,{vec {z}},,}}^{operatorname {T} },M,{vec {w}},} where {displaystyle 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 See also: Complexification Assume that {displaystyle H} is a complex Hilbert space with inner product {displaystyle langle ,cdot mid cdot ,rangle .} When the Hilbert space {displaystyle H} is reinterpreted as a real Hilbert space then it will be denoted by {displaystyle H_{mathbb {R} },} where the (real) inner-product on {displaystyle H_{mathbb {R} }} is the real part of {displaystyle H} 's inner product; that is: {displaystyle langle x,yrangle _{mathbb {R} }:=operatorname {re} langle x,yrangle .} The norm on {displaystyle H_{mathbb {R} }} induced by {displaystyle langle ,cdot ,,,cdot ,rangle _{mathbb {R} }} is equal to the original norm on {displaystyle H} and the continuous dual space of {displaystyle H_{mathbb {R} }} is the set of all real-valued bounded {displaystyle mathbb {R} } -linear functionals on {displaystyle H_{mathbb {R} }} (see the article about the polarization identity for additional details about this relationship). Let {displaystyle psi _{mathbb {R} }:=operatorname {re} psi } and {displaystyle psi _{i}:=operatorname {im} psi } denote the real and imaginary parts of a linear functional {displaystyle psi ,} so that {displaystyle psi =operatorname {re} psi +ioperatorname {im} psi =psi _{mathbb {R} }+ipsi _{i}.} The formula expressing a linear functional in terms of its real part is {displaystyle psi (h)=psi _{mathbb {R} }(h)-ipsi _{mathbb {R} }(ih)quad {text{ for }}hin H,} where {displaystyle psi _{i}(h)=-ipsi _{mathbb {R} }(ih)} for all {displaystyle hin H.} It follows that {displaystyle ker psi _{mathbb {R} }=psi ^{-1}(imathbb {R} ),} and that {displaystyle psi =0} if and only if {displaystyle psi _{mathbb {R} }=0.} It can also be shown that {displaystyle |psi |=left|psi _{mathbb {R} }right|=left|psi _{i}right|} where {displaystyle left|psi _{mathbb {R} }right|:=sup _{|h|leq 1}left|psi _{mathbb {R} }(h)right|} and {displaystyle left|psi _{i}right|:=sup _{|h|leq 1}left|psi _{i}(h)right|} are the usual operator norms. In particular, a linear functional {displaystyle psi } is bounded if and only if its real part {displaystyle psi _{mathbb {R} }} is bounded.

Representing a functional and its real part The Riesz representation of a continuous linear function {displaystyle varphi } on a complex Hilbert space is equal to the Riesz representation of its real part {displaystyle operatorname {re} varphi } on its associated real Hilbert space.

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

Furthermore, if {displaystyle varphi neq 0} then {displaystyle f_{varphi }} is perpendicular to {displaystyle ker varphi _{mathbb {R} }} with respect to {displaystyle langle cdot ,cdot rangle _{mathbb {R} }} where the kernel of {displaystyle varphi } is be a proper subspace of the kernel of its real part {displaystyle varphi _{mathbb {R} }.} Assume now that {displaystyle varphi neq 0.} Then {displaystyle f_{varphi }not in ker varphi _{mathbb {R} }} because {displaystyle varphi _{mathbb {R} }left(f_{varphi }right)=varphi left(f_{varphi }right)=|varphi |^{2}neq 0} and {displaystyle ker varphi } is a proper subset of {displaystyle ker varphi _{mathbb {R} }.} The vector subspace {displaystyle ker varphi } has real codimension {displaystyle 1} in {displaystyle ker varphi _{mathbb {R} },} while {displaystyle ker varphi _{mathbb {R} }} has real codimension {displaystyle 1} in {displaystyle H_{mathbb {R} },} and {displaystyle leftlangle f_{varphi },ker varphi _{mathbb {R} }rightrangle _{mathbb {R} }=0.} That is, {displaystyle f_{varphi }} is perpendicular to {displaystyle ker varphi _{mathbb {R} }} with respect to {displaystyle langle cdot ,cdot rangle _{mathbb {R} }.} Canonical injections into the dual and anti-dual Induced linear map into anti-dual The map defined by placing {displaystyle y} into the linear coordinate of the inner product and letting the variable {displaystyle hin H} vary over the antilinear coordinate results in an antilinear functional: {displaystyle langle ,cdot mid y,rangle =langle ,y,cdot ,rangle :Hto mathbb {F} quad {text{ defined by }}quad hmapsto langle ,hmid y,rangle =langle ,y,h,rangle .} This map is an element of {displaystyle {overline {H}}^{*},} which is the continuous anti-dual space of {displaystyle H.} The canonical map from {displaystyle H} into its anti-dual {displaystyle {overline {H}}^{*}} [1] 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}}} which is also an injective isometry.[1] 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} can be written (uniquely) in this form.[1] If {displaystyle operatorname {Cong} :H^{*}to {overline {H}}^{*}} is the canonical antilinear bijective isometry {displaystyle fmapsto {overline {f}}} that was defined above, then the following equality holds: {displaystyle operatorname {Cong} ~circ ~operatorname {In} _{H}^{H^{*}}~=~operatorname {In} _{H}^{{overline {H}}^{*}}.} Extending the bra–ket notation to bras and kets Main article: Bra–ket notation Let {displaystyle left(H,langle cdot ,cdot rangle _{H}right)} be a Hilbert space and as before, let {displaystyle langle y,|,xrangle _{H}:=langle x,yrangle _{H}.} Let {displaystyle {begin{alignedat}{4}Phi :;&&H&&;to ;&H^{*}\[0.3ex]&&g&&;mapsto ;&leftlangle ,gmid cdot ,rightrangle _{H}=leftlangle ,cdot ,g,rightrangle _{H}\end{alignedat}}} which is a bijective antilinear isometry that satisfies {displaystyle (Phi h)g=langle hmid grangle _{H}=langle g,hrangle _{H}quad {text{ for all }}g,hin H.} Bras Given a vector {displaystyle hin H,} let {displaystyle langle h,|} denote the continuous linear functional {displaystyle Phi h} ; that is, {displaystyle langle h,|~:=~Phi h} so that this functional {displaystyle langle h,|} is defined by {displaystyle gmapsto leftlangle ,hmid g,rightrangle _{H}.} This map was denoted by {displaystyle leftlangle hmid cdot ,rightrangle } earlier in this article.

The assignment {displaystyle hmapsto langle h|} is just the isometric antilinear isomorphism {displaystyle Phi ~:~Hto H^{*},} which is why {displaystyle ~langle cg+h,|~=~{overline {c}}langle gmid ~+~langle h,|~} holds for all {displaystyle g,hin H} and all scalars {displaystyle c.} The result of plugging some given {displaystyle gin H} into the functional {displaystyle langle h,|} is the scalar {displaystyle langle h,|,grangle _{H}=langle g,hrangle _{H},} which may be denoted by {displaystyle langle hmid grangle .} [note 6] Bra of a linear functional Given a continuous linear functional {displaystyle psi in H^{*},} let {displaystyle langle psi mid } denote the vector {displaystyle Phi ^{-1}psi in H} ; that is, {displaystyle langle psi mid ~:=~Phi ^{-1}psi .} The assignment {displaystyle psi mapsto langle psi mid } is just the isometric antilinear isomorphism {displaystyle Phi ^{-1}~:~H^{*}to H,} which is why {displaystyle ~langle cpsi +phi mid ~=~{overline {c}}langle psi mid ~+~langle phi mid ~} holds for all {displaystyle phi ,psi in H^{*}} and all scalars {displaystyle c.} The defining condition of the vector {displaystyle langle psi |in H} is the technically correct but unsightly equality {displaystyle leftlangle ,langle psi mid ,mid grightrangle _{H}~=~psi gquad {text{ for all }}gin H,} which is why the notation {displaystyle leftlangle psi mid grightrangle } is used in place of {displaystyle leftlangle ,langle psi mid ,mid grightrangle _{H}=leftlangle g,,langle psi mid rightrangle _{H}.} With this notation, the defining condition becomes {displaystyle leftlangle psi mid grightrangle ~=~psi gquad {text{ for all }}gin H.} Kets For any given vector {displaystyle gin H,} the notation {displaystyle |,grangle } is used to denote {displaystyle g} ; that is, {displaystyle mid grangle :=g.} The assignment {displaystyle gmapsto |,grangle } is just the identity map {displaystyle operatorname {Id} _{H}:Hto H,} which is why {displaystyle ~mid cg+hrangle ~=~cmid grangle ~+~mid hrangle ~} holds for all {displaystyle g,hin H} and all scalars {displaystyle c.} The notation {displaystyle langle hmid grangle } and {displaystyle langle psi mid grangle } is used in place of {displaystyle leftlangle hmid ,mid grangle ,rightrangle _{H}~=~leftlangle mid grangle ,hrightrangle _{H}} and {displaystyle leftlangle psi mid ,mid grangle ,rightrangle _{H}~=~leftlangle g,,langle psi mid rightrangle _{H},} respectively. As expected, {displaystyle ~langle psi mid grangle =psi g~} and {displaystyle ~langle hmid grangle ~} really is just the scalar {displaystyle ~langle hmid grangle _{H}~=~langle g,hrangle _{H}.} Adjoints and transposes Let {displaystyle A:Hto Z} be a continuous linear operator between Hilbert spaces {displaystyle left(H,langle cdot ,cdot rangle _{H}right)} and {displaystyle left(Z,langle cdot ,cdot rangle _{Z}right).} As before, let {displaystyle langle ymid xrangle _{H}:=langle x,yrangle _{H}} and {displaystyle langle ymid xrangle _{Z}:=langle x,yrangle _{Z}.} Denote by {displaystyle {begin{alignedat}{4}Phi _{H}:;&&H&&;to ;&H^{*}\[0.3ex]&&g&&;mapsto ;&langle ,gmid cdot ,rangle _{H}\end{alignedat}}quad {text{ and }}quad {begin{alignedat}{4}Phi _{Z}:;&&Z&&;to ;&Z^{*}\[0.3ex]&&y&&;mapsto ;&langle ,ymid cdot ,rangle _{Z}\end{alignedat}}} the usual bijective antilinear isometries that satisfy: {displaystyle left(Phi _{H}gright)h=langle gmid hrangle _{H}quad {text{ for all }}g,hin Hqquad {text{ and }}qquad left(Phi _{Z}yright)z=langle ymid zrangle _{Z}quad {text{ for all }}y,zin Z.} Definition of the adjoint Main articles: Hermitian adjoint and Conjugate transpose For every {displaystyle zin Z,} the scalar-valued map {displaystyle langle zmid A(cdot )rangle _{Z}} [note 7] on {displaystyle H} defined by {displaystyle hmapsto langle zmid Ahrangle _{Z}=langle Ah,zrangle _{Z}} is a continuous linear functional on {displaystyle H} and so by the Riesz representation theorem, there exists a unique vector in {displaystyle H,} denoted by {displaystyle A^{*}z,} such that {displaystyle langle zmid A(cdot )rangle _{Z}=leftlangle A^{*}zmid cdot ,rightrangle _{H},} or equivalently, such that {displaystyle langle zmid Ahrangle _{Z}=leftlangle A^{*}zmid hrightrangle _{H}quad {text{ for all }}hin H.} The assignment {displaystyle zmapsto A^{*}z} thus induces a function {displaystyle A^{*}:Zto H} called the adjoint of {displaystyle A:Hto Z} whose defining condition is {displaystyle langle zmid Ahrangle _{Z}=leftlangle A^{*}zmid hrightrangle _{H}quad {text{ for all }}hin H{text{ and all }}zin Z.} The adjoint {displaystyle A^{*}:Zto H} is necessarily a continuous (equivalently, a bounded) linear operator.

If {displaystyle H} is finite dimensional with the standard inner product and if {displaystyle M} is the transformation matrix of {displaystyle A} with respect to the standard orthonormal basis then {displaystyle M} 's conjugate transpose {displaystyle {overline {M^{operatorname {T} }}}} is the transformation matrix of the adjoint {displaystyle A^{*}.} Adjoints are transposes Main article: Transpose of a linear map See also: Transpose It is also possible to define the transpose or algebraic adjoint of {displaystyle A:Hto Z,} which is the map {displaystyle {}^{t}A:Z^{*}to H^{*}} defined by sending a continuous linear functionals {displaystyle psi in Z^{*}} to {displaystyle {}^{t}A(psi ):=psi circ A,} where {displaystyle psi circ A} is always a continuous linear functional on {displaystyle H.} It satisfies {displaystyle |A|=left|{}^{t}Aright|} (this is true more generally, when {displaystyle H} and {displaystyle Z} are merely normed spaces).[4] The adjoint {displaystyle A^{*}:Zto H} is actually just to the transpose {displaystyle {}^{t}A:Z^{*}to H^{*}} [2] when the Riesz representation theorem is used to identify {displaystyle Z} with {displaystyle Z^{*}} and {displaystyle H} with {displaystyle H^{*}.} Explicitly, the relationship between the adjoint and transpose is: {displaystyle {}^{t}A~circ ~Phi _{Z}~=~Phi _{H}~circ ~A^{*}}         (Adjoint-transpose) show Proof This 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}.} Given any {displaystyle zin Z,} the left and right hand sides of equality (Adjoint-transpose) can be rewritten in terms of the inner products: {displaystyle left({}^{t}A~circ ~Phi _{Z}right)z=langle zmid A(cdot )rangle _{Z}quad {text{ and }}quad left(Phi _{H}~circ ~A^{*}right)z=langle A^{*}zmid cdot ,rangle _{H}} where as before, {displaystyle langle zmid A(cdot )rangle _{Z}} denotes the continuous linear functional on {displaystyle H} defined by {displaystyle gmapsto langle zmid Agrangle _{Z}.} [note 7] Descriptions of self-adjoint, normal, and unitary operators Assume {displaystyle Z=H} and let {displaystyle Phi :=Phi _{H}=Phi _{Z}.} Let {displaystyle A:Hto H} be a continuous (that is, bounded) linear operator.

Whether or not {displaystyle A:Hto H} is self-adjoint, normal, or unitary depends entirely on whether or not {displaystyle A} satisfies certain defining conditions related to its adjoint, which was shown by (Adjoint-transpose) to essentially be just the transpose {displaystyle {}^{t}A:H^{*}to H^{*}.} Because the transpose of {displaystyle A} 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 {displaystyle H} that can be defined entirely in terms of {displaystyle A,} the inner product {displaystyle langle ,cdot mid cdot ,rangle } on {displaystyle H,} and some given vector {displaystyle hin H.} These "elementary {displaystyle A} -induced" continuous linear functionals are {displaystyle leftlangle Ahmid cdot ,rightrangle } and {displaystyle langle hmid A(cdot )rangle } [note 7] where {displaystyle leftlangle Ahmid cdot ,rightrangle =Phi (Ah)=(Phi circ A)hquad {text{ and }}quad langle hmid A(cdot )rangle =left({}^{t}Acirc Phi right)h.} Self-adjoint operators See also: Self-adjoint operator, Hermitian matrix, and Symmetric matrix A continuous linear operator {displaystyle A:Hto H} is called self-adjoint it is equal to its own adjoint; that is, if {displaystyle A=A^{*}.} Using (Adjoint-transpose), this happens if and only if: {displaystyle Phi circ A={}^{t}Acirc Phi } where this equality can be rewritten in the following two equivalent forms: {displaystyle A=Phi ^{-1}circ {}^{t}Acirc Phi quad {text{ or }}quad {}^{t}A=Phi circ Acirc Phi ^{-1}.} Unraveling notation and definitions produces the following characterization of self-adjoint operators in terms of the aforementioned " {displaystyle A} -induced" continuous linear functionals: {displaystyle A} is self-adjoint if and only if for all {displaystyle zin H,} the linear functional {displaystyle langle zmid A(cdot )rangle } [note 7] is equal to the linear functional {displaystyle langle Azmid cdot ,rangle } ; that is, if and only if {displaystyle langle Azmid cdot ,rangle =langle zmid A(cdot )rangle quad {text{ for all }}zin H.}         (Self-adjointness functionals) Normal operators See also: Normal operator and Normal matrix A continuous linear operator {displaystyle A:Hto H} is called normal if {displaystyle AA^{*}=A^{*}A,} which happens if and only if for all {displaystyle z,hin H,} {displaystyle leftlangle AA^{*}zmid hrightrangle =leftlangle A^{*}Azmid hrightrangle .} 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: {displaystyle A} is a normal operator if and only if {displaystyle leftlangle ,langle Ahmid cdot ,rangle mid langle Azmid cdot ,rangle ,rightrangle _{H^{*}}~=~leftlangle ,langle h|A(cdot )rangle mid langle zmid A(cdot )rangle ,rightrangle _{H^{*}}quad {text{ for all }}z,hin H}         (Normality functionals) where the left hand side is also equal to {displaystyle {overline {langle Ahmid Azrangle }}_{H}=langle Azmid Ahrangle _{H}.} The left hand side of this characterization involves only linear functionals of the form {displaystyle langle Ahmid cdot ,rangle } while the right hand side involves only linear functions of the form {displaystyle langle hmid A(cdot )rangle } (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 {displaystyle z,hin H} for both forms). In other words, if it happens to be the case that the assignment of linear functionals {displaystyle langle Ahmid cdot ,rangle ~mapsto ~langle h|A(cdot )rangle } is well-defined (or alternatively, if {displaystyle langle h|A(cdot )rangle ~mapsto ~langle Ahmid cdot ,rangle } is well-defined) where {displaystyle h} ranges over {displaystyle H} (which happens if {displaystyle A} is injective, for instance), then {displaystyle A} is a normal operator if and only if this assignment preserves the inner product on {displaystyle H^{*}.} The fact that every self-adjoint bounded linear operator is normal follows readily by direct substitution of {displaystyle A^{*}=A} into either side of {displaystyle A^{*}A=AA^{*}.} 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 {displaystyle A} is a normal operator if and only if {displaystyle |Az|=left|A^{*}zright|} for every {displaystyle zin H,} [2] which happens if and only if {displaystyle |Az|_{H}=|langle z,|,A(cdot )rangle |_{H^{*}}quad {text{ for every }}zin H.} Unitary operators See also: Unitary transformation and Unitary matrix An invertible bounded linear operator {displaystyle A:Hto H} is said to be unitary if its inverse is its adjoint: {displaystyle A^{-1}=A^{*}.} By using (Adjoint-transpose), this is seen to be equivalent to {displaystyle Phi circ A^{-1}={}^{t}Acirc Phi .} Unraveling notation and definitions, it follows that {displaystyle A} is unitary if and only if {displaystyle langle A^{-1}zmid cdot ,rangle =langle zmid A(cdot )rangle quad {text{ for all }}zin H.} The fact that a bounded invertible linear operator {displaystyle A:Hto H} is unitary if and only if {displaystyle A^{*}A=operatorname {Id} _{H}} (or equivalently, {displaystyle {}^{t}Acirc Phi circ A=Phi } ) produces another (well-known) characterization: an invertible bounded linear map {displaystyle A} is unitary if and only if {displaystyle langle Azmid A(cdot ),rangle =langle zmid cdot ,rangle quad {text{ for all }}zin H.} Because {displaystyle A:Hto H} is invertible (and so in particular a bijection), this is also true of the transpose {displaystyle {}^{t}A:H^{*}to H^{*}.} This fact also allows the vector {displaystyle zin H} in the above characterizations to be replaced with {displaystyle Az} or {displaystyle A^{-1}z,} thereby producing many more equalities. Similarly, {displaystyle ,cdot ,} can be replaced with {displaystyle A(cdot )} or {displaystyle A^{-1}(cdot ).} See also Choquet theory Fundamental theorem of Hilbert spaces Matrix coefficient – Functions on special groups related to their matrix representations Citations ^ Jump up to: a b c d e f g h i j k l Trèves 2006, pp. 112–123. ^ Jump up to: a b c Rudin 1991, pp. 306–312. ^ Rudin 1991, pp. 307−309. ^ Rudin 1991, pp. 92–115. Notes ^ If {displaystyle mathbb {F} =mathbb {R} } then the inner product will be symmetric so it does not matter which coordinate of the inner product the element {displaystyle y} is placed into because the same map will result. But if {displaystyle mathbb {F} =mathbb {C} } then except for the constant {displaystyle 0} map, antilinear functionals on {displaystyle H} are completely distinct from linear functionals on {displaystyle H,} which makes the coordinate that {displaystyle y} is placed into is very important. For a non-zero {displaystyle yin H} to induce a linear functional (rather than an antilinear functional), {displaystyle y} 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 {displaystyle hmapsto langle y,hrangle =langle hmid yrangle ,} which is not a linear functional on {displaystyle H} and so it will not be an element of the continuous dual space {displaystyle H^{*}.} ^ This means that for all vectors {displaystyle yin H:} (1) {displaystyle Phi :Hto H^{*}} is injective. (2) The norms of {displaystyle y} and {displaystyle Phi (y)} are the same: {displaystyle |Phi (y)|=|y|.} (3) {displaystyle Phi } is an additive map, meaning that {displaystyle Phi (x+y)=Phi (x)+Phi (y)} for all {displaystyle x,yin H.} (4) {displaystyle Phi } is conjugate homogeneous: {displaystyle Phi (sy)={overline {s}}Phi (y)} for all scalars {displaystyle s.} (5) {displaystyle Phi } is real homogeneous: {displaystyle Phi (ry)=rPhi (y)} for all real numbers {displaystyle rin mathbb {R} .} ^ Jump up to: a b This footnote explains how to define - using only {displaystyle H} 's operations - addition and scalar multiplication of affine hyperplanes so that these operations correspond to addition and scalar multiplication of linear functionals. Let {displaystyle H} be any vector space and let {displaystyle H^{#}} denote its algebraic dual space. Let {displaystyle {mathcal {A}}:=left{varphi ^{-1}(1):varphi in H^{#}right}} and let {displaystyle ,{hat {cdot }},} and {displaystyle ,{hat {+}},} denote the (unique) vector space operations on {displaystyle {mathcal {A}}} that make the bijection {displaystyle I:H^{#}to {mathcal {A}}} defined by {displaystyle varphi mapsto varphi ^{-1}(1)} into a vector space isomorphism. Note that {displaystyle varphi ^{-1}(1)=varnothing } if and only if {displaystyle varphi =0,} so {displaystyle varnothing } is the additive identity of {displaystyle left({mathcal {A}},{hat {+}},{hat {cdot }}right)} (because this is true of {displaystyle I^{-1}(varnothing )=0} in {displaystyle H^{#}} and {displaystyle I} is a vector space isomorphism). For every {displaystyle Ain {mathcal {A}},} let {displaystyle ker A=H} if {displaystyle A=varnothing } and let {displaystyle ker A=A-A} otherwise; if {displaystyle A=I(varphi )=varphi ^{-1}(1)} then {displaystyle ker A=ker varphi } so this definition is consistent with the usual definition of the kernel of a linear functional. Say that {displaystyle A,Bin {mathcal {A}}} are parallel if {displaystyle ker A=ker B,} where if {displaystyle A} and {displaystyle B} are not empty then this happens if and only if the linear functionals {displaystyle I^{-1}(A)} and {displaystyle I^{-1}(B)} are non-zero scalar multiples of each other. The vector space operations on the vector space of affine hyperplanes {displaystyle {mathcal {A}}} are now described in a way that involves only the vector space operations on {displaystyle H} ; this results in an interpretation of the vector space operations on the algebraic dual space {displaystyle H^{#}} that is entirely in terms of affine hyperplanes. Fix hyperplanes {displaystyle A,Bin {mathcal {A}}.} If {displaystyle s} is a scalar then {displaystyle s{hat {cdot }}A=left{hin H:shin Aright}.} Describing the operation {displaystyle A{hat {+}}B} in terms of only the sets {displaystyle A=varphi ^{-1}(1)} and {displaystyle B=psi ^{-1}(1)} is more complicated because by definition, {displaystyle A{hat {+}}B=I(varphi ){hat {+}}I(psi ):=I(varphi +psi )=(varphi +psi )^{-1}(1).} If {displaystyle A=varnothing } (respectively, if {displaystyle B=varnothing } ) then {displaystyle A{hat {+}}B} is equal to {displaystyle B} (resp. is equal to {displaystyle A} ) so assume {displaystyle Aneq varnothing } and {displaystyle Bneq varnothing .} The hyperplanes {displaystyle A} and {displaystyle B} are parallel if and only if there exists some scalar {displaystyle r} (necessarily non-0) such that {displaystyle A=rB,} in which case {displaystyle A{hat {+}}B=left{hin H:(1+r)hin Bright};} this can optionally be subdivided into two cases: if {displaystyle r=-1} (which happens if and only if the linear functionals {displaystyle I^{-1}(A)} and {displaystyle I^{-1}(B)} are negatives of each) then {displaystyle A{hat {+}}B=varnothing } while if {displaystyle rneq -1} then {displaystyle A{hat {+}}B={frac {1}{1+r}}B={frac {r}{1+r}}A.} Finally, assume now that {displaystyle ker Aneq ker B.} Then {displaystyle A{hat {+}}B} is the unique affine hyperplane containing both {displaystyle Acap ker B} and {displaystyle Bcap ker A} as subsets; explicitly, {displaystyle ker left(A{hat {+}}Bright)=operatorname {span} left(Acap ker B-Bcap ker Aright)} and {displaystyle A{hat {+}}B=Acap ker B+ker left(A{hat {+}}Bright)=Bcap ker A+ker left(A{hat {+}}Bright).} To see why this formula for {displaystyle A{hat {+}}B} should hold, consider {displaystyle H:=mathbb {R} ^{3},} {displaystyle A:=varphi ^{-1}(1),} and {displaystyle B:=psi ^{-1}(1),} where {displaystyle varphi (x,y,z):=x} and {displaystyle psi (x,y,z):=x+y} (or alternatively, {displaystyle psi (x,y,z):=y} ). Then by definition, {displaystyle A{hat {+}}B:=(varphi +psi )^{-1}(1)} and {displaystyle ker left(A{hat {+}}Bright):=(varphi +psi )^{-1}(0).} Now {displaystyle Acap ker B~=~varphi ^{-1}(1)cap psi ^{-1}(0)~subseteq ~(varphi +psi )^{-1}(1)} is an affine subspace of codimension {displaystyle 2} in {displaystyle H} (it is equal to a translation of the {displaystyle z} -axis {displaystyle {(0,0)}times mathbb {R} } ). The same is true of {displaystyle Bcap ker A.} Plotting an {displaystyle x} - {displaystyle y} -plane cross section (that is, setting {displaystyle z=} constant) of the sets {displaystyle ker A,ker B,A} and {displaystyle B} (each of which will be plotted as a line), the set {displaystyle (varphi +psi )^{-1}(1)} will then be plotted as the (unique) line passing through the {displaystyle Acap ker B} and {displaystyle Bcap ker A} (which will be plotted as two distinct points) while {displaystyle (varphi +psi )^{-1}(0)=ker left(A{hat {+}}Bright)} will be plotted the line through the origin that is parallel to {displaystyle A{hat {+}}B=(varphi +psi )^{-1}(1).} The above formulas for {displaystyle ker left(A{hat {+}}Bright):=(varphi +psi )^{-1}(0)} and {displaystyle A{hat {+}}B:=(varphi +psi )^{-1}(1)} follow naturally from the plot and they also hold in general. ^ Showing that there is a non-zero vector {displaystyle v} in {displaystyle K^{bot }} relies on the continuity of {displaystyle phi } and the Cauchy completeness of {displaystyle H.} This is the only place in the proof in which these properties are used. ^ Technically, {displaystyle H=Koplus K^{bot }} means that the addition map {displaystyle Ktimes K^{bot }to H} defined by {displaystyle (k,p)mapsto k+p} is a surjective linear isomorphism and homeomorphism. See the article on complemented subspaces for more details. ^ The usual notation for plugging an element {displaystyle g} into a linear map {displaystyle F} is {displaystyle F(g)} and sometimes {displaystyle Fg.} Replacing {displaystyle F} with {displaystyle langle hmid :=~Phi h} produces {displaystyle langle hmid (g)} or {displaystyle langle hmid g,} which is unsightly (despite being consistent with the usual notation used with functions). Consequently, the symbol {displaystyle ,rangle ,} is appended to the end, so that the notation {displaystyle langle hmid grangle } is used instead to denote this value {displaystyle (Phi h)g.} ^ Jump up to: a b c d e The notation {displaystyle leftlangle zmid A(cdot )rightrangle _{Z}} denotes the continuous linear functional defined by {displaystyle gmapsto leftlangle zmid Agrightrangle _{Z}.} Proofs ^ This is because {displaystyle x_{K}=x-{frac {leftlangle x,f_{varphi }rightrangle }{left|f_{varphi }right|^{2}}}f_{varphi }.} Now use {displaystyle left|f_{varphi }right|^{2}=|varphi |^{2}} and {displaystyle leftlangle x,f_{varphi }rightrangle =varphi (x)} and solve for {displaystyle f_{varphi }.} {displaystyle blacksquare } ^ {displaystyle leftlangle A^{*}Azmid hrightrangle =leftlangle ,Azmid Ah,rightrangle _{H}=leftlangle ,Phi Ahmid Phi Az,rightrangle _{H^{*}}} where {displaystyle Phi Ah:=leftlangle Ahmid cdot ,rightrangle } and {displaystyle Phi Az:=leftlangle Azmid cdot ,rightrangle .} By definition of the adjoint, {displaystyle leftlangle A^{*}hmid A^{*}z,rightrangle =leftlangle hmid AA^{*}z,rightrangle } so taking the complex conjugate of both sides proves that {displaystyle leftlangle AA^{*}zmid hrightrangle =leftlangle A^{*}zmid A^{*}hrightrangle .} From {displaystyle A^{*}=Phi ^{-1}circ {}^{t}Acirc Phi ,} it follows that {displaystyle leftlangle AA^{*}z,|,hrightrangle _{H}=leftlangle A^{*}zmid A^{*}hrightrangle _{H}=leftlangle Phi ^{-1}circ {}^{t}Acirc Phi zmid Phi ^{-1}circ {}^{t}Acirc Phi hrightrangle _{H}=leftlangle ,{}^{t}Acirc Phi hmid {}^{t}Acirc Phi zrightrangle _{H^{*}}} where {displaystyle left({}^{t}Acirc Phi right)h=langle h,|,A(cdot )rangle } and {displaystyle left({}^{t}Acirc Phi right)z=langle z,|,A(cdot )rangle .} {displaystyle blacksquare } 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. 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. show vte Functional analysis (topics – glossary) show vte Hilbert spaces Categories: Theorems in functional analysisDuality theoriesIntegral representations