TheRiesz representation theorem, sometimes called theRiesz–Fréchet representation theorem afterFrigyes Riesz andMaurice René Fréchet, establishes an important connection between aHilbert space and itscontinuous dual space. If the underlyingfield is thereal numbers, the two areisometricallyisomorphic; if the underlying field is thecomplex numbers, the two are isometricallyanti-isomorphic. The (anti-)isomorphism is a particularnatural isomorphism.

Let${\displaystyle H}$ be aHilbert 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 acomplex Hilbert space (resp. areal Hilbert space). Every real Hilbert space can be extended to be adense subset of a unique (up tobijectiveisometry) complex Hilbert space, called itscomplexification, 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 bothmathematicians andphysicists 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 realor complex Hilbert space.

By definition, anantilinear map (also called aconjugate-linear map)${\displaystyle f:H\to Y}$ is a map betweenvector spaces that isadditive:$${\displaystyle f(x+y)=f(x)+f(y)\text{ for all }x,y\in H,}$$andantilinear (also calledconjugate-linear orconjugate-homogeneous):$${\displaystyle f(cx)=\overline{c}f(x)\text{ for all }x\in H\text{ and all scalar }c\in \mathbb{F},}$$where${\displaystyle \overline{c}}$ is the conjugate of the complex number${\displaystyle c=a+bi}$, given by${\displaystyle \overline{c}=a-bi}$.

In contrast, a map${\displaystyle f:H\to Y}$ islinear if it is additive andhomogeneous:$${\displaystyle f(cx)=cf(x)\text{ for all }x\in H\text{ and all scalars }c\in \mathbb{F}.}$$

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 aBanach space (or more generally, from any Banach space into anytopological vector space) iscontinuousif and only if it isbounded; the same is true of antilinear maps. Theinverse of any antilinear (resp. linear) bijection is again an antilinear (resp. linear) bijection. The composition of twoantilinear maps is alinear map.

Continuous dual and anti-dual spaces

Afunctional on${\displaystyle H}$ is a function${\displaystyle H\to \mathbb{F}}$ whosecodomain is the underlying scalar field${\displaystyle \mathbb{F}.}$ Denote by${\displaystyle H^{\ast }}$ (resp. by${\displaystyle {\overline{H}}^{\ast })}$ 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,${\displaystyle H^{\ast }={\overline{H}}^{\ast }.}$

One-to-one correspondence between linear and antilinear functionals

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

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${\displaystyle f\mapsto \overline{f}}$ reduces down to theidentity map.

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

all functionals (resp. all linear functionals, all continuous linear functionals${\displaystyle H^{\ast }}$) on${\displaystyle H,}$

onto the set of

all functionals (resp. allantilinear functionals, all continuousantilinear functionals${\displaystyle {\overline{H}}^{\ast }}$) on${\displaystyle H.}$

TheHilbert space${\displaystyle H}$ has an associatedinner product${\displaystyle H\times H\to \mathbb{F}}$ valued in${\displaystyle H}$'s underlying scalar field${\displaystyle \mathbb{F}}$ that is linear in one coordinate and antilinear in the other (as specified below).If${\displaystyle H}$ is a complex Hilbert space (${\displaystyle \mathbb{F}=\mathbb{C}}$), then there is a crucial difference between the notations prevailing in mathematics versus physics, regarding which of the two variables is linear.However, for real Hilbert spaces (${\displaystyle \mathbb{F}=\mathbb{R}}$), the inner product is asymmetric map that is linear in each coordinate (bilinear), so there can be no such confusion.

Inmathematics, the inner product on a Hilbert space${\displaystyle H}$ is often denoted by${\displaystyle ⟨\cdot ,\cdot ⟩}$ or${\displaystyle {⟨\cdot ,\cdot ⟩}_H}$ while inphysics, thebra–ket notation${\displaystyle ⟨\cdot \mid \cdot ⟩}$ or${\displaystyle {⟨\cdot \mid \cdot ⟩}_H}$ is typically used. In this article, these two notations will be related by the equality:

$${\displaystyle ⟨x,y⟩:=⟨y\mid x⟩\text{ for all }x,y\in H.}$$These have the following properties:

1. The map${\displaystyle ⟨\cdot ,\cdot ⟩}$ islinear in its first coordinate; equivalently, the map${\displaystyle ⟨\cdot \mid \cdot ⟩}$ islinear in its second coordinate. That is, for fixed${\displaystyle y\in H,}$ the map${\displaystyle ⟨y\mid \cdot ⟩=⟨\cdot ,y⟩:H\to \mathbb{F}}$ with$h\mapsto ⟨y\mid h⟩=⟨h,y⟩$is a linear functional on${\displaystyle H.}$ This linear functional is continuous, so${\displaystyle ⟨y\mid \cdot ⟩=⟨\cdot ,y⟩\in H^{\ast }.}$
2. The map${\displaystyle ⟨\cdot ,\cdot ⟩}$ isantilinear in itssecond coordinate; equivalently, the map${\displaystyle ⟨\cdot \mid \cdot ⟩}$ isantilinear in itsfirst coordinate. That is, for fixed${\displaystyle y\in H,}$ the map${\displaystyle ⟨\cdot \mid y⟩=⟨y,\cdot ⟩:H\to \mathbb{F}}$ with$h\mapsto ⟨h\mid y⟩=⟨y,h⟩$is an antilinear functional on${\displaystyle H.}$ This antilinear functional is continuous, so${\displaystyle ⟨\cdot \mid y⟩=⟨y,\cdot ⟩\in {\overline{H}}^{\ast }.}$

In computations, one must consistently use either the mathematics notation${\displaystyle ⟨\cdot ,\cdot ⟩}$, which is (linear, antilinear); or the physics notation${\displaystyle ⟨\cdot \mid \cdot ⟩}$, which is (antilinear | linear).

If${\displaystyle x=y}$ then${\displaystyle ⟨x\mid x⟩=⟨x,x⟩}$ is a non-negative real number and the map$${\displaystyle \Vert x\Vert :=\sqrt{⟨x,x⟩}=\sqrt{⟨x\mid x⟩}}$$

defines acanonical norm on${\displaystyle H}$ that makes${\displaystyle H}$ into anormed space.^[1] As with all normed spaces, the (continuous) dual space${\displaystyle H^{\ast }}$ carries a canonical norm, called thedual norm, that is defined by^[1]$${\displaystyle \Vert f{\Vert }_{H^{\ast }}:=\underset{\Vert x\Vert \le 1,x\in H}{sup}|f(x)|\text{ for every }f\in H^{\ast }.}$$

The canonical norm on the (continuous)anti-dual space${\displaystyle {\overline{H}}^{\ast },}$ denoted by${\displaystyle \Vert f{\Vert }_{{\overline{H}}^{\ast }},}$ is defined by using this same equation:^[1]$${\displaystyle \Vert f{\Vert }_{{\overline{H}}^{\ast }}:=\underset{\Vert x\Vert \le 1,x\in H}{sup}|f(x)|\text{ for every }f\in {\overline{H}}^{\ast }.}$$

This canonical norm on${\displaystyle H^{\ast }}$ satisfies theparallelogram law, which means that thepolarization identity can be used to define acanonical inner product on${\displaystyle H^{\ast },}$ which this article will denote by the notations$${\displaystyle {⟨f,g⟩}_{H^{\ast }}:={⟨g\mid f⟩}_{H^{\ast }},}$$ where this inner product turns${\displaystyle H^{\ast }}$ into a Hilbert space. There are now two ways of defining a norm on${\displaystyle H^{\ast }:}$ the norm induced by this inner product (that is, the norm defined by${\displaystyle f\mapsto \sqrt{{⟨f,f⟩}_{H^{\ast }}}}$) and the usualdual norm (defined as the supremum over the closedunit ball). These norms are the same; explicitly, this means that the following holds for every${\displaystyle f\in H^{\ast }:}$$${\displaystyle \underset{\Vert x\Vert \le 1,x\in H}{sup}|f(x)|=\Vert f{\Vert }_{H^{\ast }}=\sqrt{⟨f,f{⟩}_{H^{\ast }}}=\sqrt{⟨f\mid f{⟩}_{H^{\ast }}}.}$$

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${\displaystyle H^{\ast }.}$

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

Canonical isometry between the dual and antidual

Thecomplex conjugate${\displaystyle \overline{f}}$ of a functional${\displaystyle f,}$ which was defined above, satisfies$${\displaystyle \Vert f{\Vert }_{H^{\ast }}={\Vert \overline{f}\Vert }_{{\overline{H}}^{\ast }}\text{ and }{\Vert \overline{g}\Vert }_{H^{\ast }}=\Vert g{\Vert }_{{\overline{H}}^{\ast }}}$$for every${\displaystyle f\in H^{\ast }}$ and every${\displaystyle g\in {\overline{H}}^{\ast }.}$ This says exactly that the canonical antilinearbijection defined byas well as its inverse${\displaystyle {\mathrm{Cong}}^{-1}:{\overline{H}}^{\ast }\to H^{\ast }}$ are antilinearisometries and consequently alsohomeomorphisms. The inner products on the dual space${\displaystyle H^{\ast }}$ and the anti-dual space${\displaystyle {\overline{H}}^{\ast },}$ denoted respectively by${\displaystyle ⟨\cdot ,\cdot {⟩}_{H^{\ast }}}$ and${\displaystyle ⟨\cdot ,\cdot {⟩}_{{\overline{H}}^{\ast }},}$ are related by$${\displaystyle ⟨\overline{f}|\overline{g}{⟩}_{{\overline{H}}^{\ast }}=\overline{⟨f|g{⟩}_{H^{\ast }}}=⟨g|f{⟩}_{H^{\ast }}\text{ for all }f,g\in H^{\ast }}$$and$${\displaystyle ⟨\overline{f}|\overline{g}{⟩}_{H^{\ast }}=\overline{⟨f|g{⟩}_{{\overline{H}}^{\ast }}}=⟨g|f{⟩}_{{\overline{H}}^{\ast }}\text{ for all }f,g\in {\overline{H}}^{\ast }.}$$

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

Two vectors${\displaystyle x}$ and${\displaystyle y}$ areorthogonal if${\displaystyle ⟨x,y⟩=0,}$ which happens if and only if${\displaystyle \Vert y\Vert \le \Vert y+sx\Vert }$ for all scalars${\displaystyle s.}$^[2] Theorthogonal complement of a subset${\displaystyle X\subseteq H}$ is$${\displaystyle X^{\mathrm{\perp }}:=\{y\in H:⟨y,x⟩=0\text{ for all }x\in X\},}$$which is always aclosed vector subspace of${\displaystyle H.}$ TheHilbert projection theorem guarantees that for anynonempty closedconvex subset${\displaystyle C}$ of aHilbert space there exists a unique vector${\displaystyle m\in C}$ such that${\displaystyle \Vert m\Vert =\underset{c\in C}{inf}\Vert c\Vert ;}$ that is,${\displaystyle m\in C}$ is the (unique)global minimum point of the function${\displaystyle C\to [0,\mathrm{\infty })}$ defined by${\displaystyle c\mapsto \Vert c\Vert .}$

Historically, the theorem is often attributed simultaneously toRiesz andFréchet in 1907 (see references).

If${\displaystyle \phi \in H^{\ast }}$ then$${\displaystyle \phi \left(f_{\phi }\right)=⟨f_{\phi },f_{\phi }⟩={\Vert f_{\phi }\Vert }^2=\Vert \phi {\Vert }^2.}$$ So in particular,${\displaystyle \phi \left(f_{\phi }\right)\ge 0}$ is always real and furthermore,${\displaystyle \phi \left(f_{\phi }\right)=0}$ if and only if${\displaystyle f_{\phi }=0}$ if and only if${\displaystyle \phi =0.}$

Linear functionals as affine hyperplanes

A non-trivial continuous linear functional${\displaystyle \phi }$ is often interpreted geometrically by identifying it with the affine hyperplane${\displaystyle A:={\phi }^{-1}(1)}$ (the kernel${\displaystyle \ker \phi ={\phi }^{-1}(0)}$ is also often visualized alongside${\displaystyle A:={\phi }^{-1}(1)}$ although knowing${\displaystyle A}$ is enough to reconstruct${\displaystyle \ker \phi }$ because if${\displaystyle A=\varnothing }$ then${\displaystyle \ker \phi =H}$ and otherwise${\displaystyle \ker \phi =A-A}$). In particular, the norm of${\displaystyle \phi }$ should somehow be interpretable as the "norm of the hyperplane${\displaystyle A}$". When${\displaystyle \phi \ne 0}$ then the Riesz representation theorem provides such an interpretation of${\displaystyle \Vert \phi \Vert }$ in terms of the affine hyperplane^{[note 3]}${\displaystyle A:={\phi }^{-1}(1)}$ as follows: using the notation from the theorem's statement, from${\displaystyle \Vert \phi {\Vert }^2\ne 0}$ it follows that${\displaystyle C:={\phi }^{-1}\left(\Vert \phi {\Vert }^2\right)=\Vert \phi {\Vert }^2{\phi }^{-1}(1)=\Vert \phi {\Vert }^{2}A}$ and so${\displaystyle \Vert \phi \Vert =\Vert f_{\phi }\Vert =\underset{c\in C}{inf}\Vert c\Vert }$ implies${\displaystyle \Vert \phi \Vert =\underset{a\in A}{inf}\Vert \phi {\Vert }^2\Vert a\Vert }$ and thus${\displaystyle \Vert \phi \Vert =\frac{1}{\underset{a\in A}{inf}\Vert a\Vert }.}$ This can also be seen by applying theHilbert projection theorem to${\displaystyle A}$ and concluding that the global minimum point of the map${\displaystyle A\to [0,\mathrm{\infty })}$ defined by${\displaystyle a\mapsto \Vert a\Vert }$ is${\displaystyle \frac{f_{\phi }}{\Vert \phi {\Vert }^2}\in A.}$ The formulas$${\displaystyle \frac{1}{\underset{a\in A}{inf}\Vert a\Vert }=\underset{a\in A}{sup}\frac{1}{\Vert a\Vert }}$$provide the promised interpretation of the linear functional's norm${\displaystyle \Vert \phi \Vert }$ entirely in terms of its associated affine hyperplane${\displaystyle A={\phi }^{-1}(1)}$ (because with this formula, knowing only theset${\displaystyle A}$ is enough to describe the norm of its associated linearfunctional). Defining${\displaystyle \frac{1}{\mathrm{\infty }}:=0,}$ theinfimum formula$${\displaystyle \Vert \phi \Vert =\frac{1}{\underset{a\in {\phi }^{-1}(1)}{inf}\Vert a\Vert }}$$will also hold when${\displaystyle \phi =0.}$ When the supremum is taken in${\displaystyle \mathbb{R}}$ (as is typically assumed), then the supremum of the empty set is${\displaystyle sup\varnothing =-\mathrm{\infty }}$ but if the supremum is taken in the non-negative reals${\displaystyle [0,\mathrm{\infty })}$ (which is theimage/range of the norm${\displaystyle \Vert \cdot \Vert }$ when${\displaystyle \dim H>0}$) then this supremum is instead${\displaystyle sup\varnothing =0,}$ in which case the supremum formula${\displaystyle \Vert \phi \Vert =\underset{a\in {\phi }^{-1}(1)}{sup}\frac{1}{\Vert a\Vert }}$ will also hold when${\displaystyle \phi =0}$ (although the atypical equality${\displaystyle sup\varnothing =0}$ is usually unexpected and so risks causing confusion).

Using the notation from the theorem above, several ways of constructing${\displaystyle f_{\phi }}$ from${\displaystyle \phi \in H^{\ast }}$ are now described. If${\displaystyle \phi =0}$ then${\displaystyle f_{\phi }:=0}$; in other words,$${\displaystyle f_0=0.}$$

This special case of${\displaystyle \phi =0}$ is henceforth assumed to be known, which is why some of the constructions given below start by assuming${\displaystyle \phi \ne 0.}$

Orthogonal complement of kernel

If${\displaystyle \phi \ne 0}$ then for any${\displaystyle 0\ne u\in (\ker \phi {)}^{\mathrm{\perp }},}$$${\displaystyle f_{\phi }:=\frac{\overline{\phi (u)}u}{\Vert u{\Vert }^2}.}$$

If${\displaystyle u\in (\ker \phi {)}^{\mathrm{\perp }}}$ is aunit vector (meaning${\displaystyle \Vert u\Vert =1}$) then$${\displaystyle f_{\phi }:=\overline{\phi (u)}u}$$(this is true even if${\displaystyle \phi =0}$ because in this case${\displaystyle f_{\phi }=\overline{\phi (u)}u=\overline{0}u=0}$). 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${\displaystyle (\ker \phi {)}^{\mathrm{\perp }}.}$ However,${\displaystyle \overline{\phi (-u)}(-u)=\overline{\phi (u)}u=f_{\phi }}$ so both these vectors result in the same${\displaystyle f_{\phi }.}$

Orthogonal projection onto kernel

If${\displaystyle x\in H}$ is such that${\displaystyle \phi (x)\ne 0}$ and if${\displaystyle x_K}$ is theorthogonal projection of${\displaystyle x}$ onto${\displaystyle \ker \phi }$ then^{[proof 1]}$${\displaystyle f_{\phi }=\frac{\Vert \phi {\Vert }^2}{\phi (x)}\left(x-x_K\right).}$$

Orthonormal basis

Given anorthonormal basis${\displaystyle {\left\{e_i\right\}}_{i\in I}}$ of${\displaystyle H}$ and a continuous linear functional${\displaystyle \phi \in H^{\ast },}$ the vector${\displaystyle f_{\phi }\in H}$ can be constructed uniquely by$${\displaystyle f_{\phi }=\sum _{i\in I}\overline{\phi \left(e_i\right)}{e}_i}$$ where all but at most countably many${\displaystyle \phi \left(e_i\right)}$ will be equal to${\displaystyle 0}$ and where the value of${\displaystyle f_{\phi }}$ 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${\displaystyle y\in H}$ is written as${\displaystyle y=\sum _{i\in I}{a}_i{e}_i}$ then$${\displaystyle \phi (y)=\sum _{i\in I}\phi \left(e_i\right)a_i=⟨f_{\phi }|y⟩}$$and$${\displaystyle {\Vert f_{\phi }\Vert }^2=\phi \left(f_{\phi }\right)=\sum _{i\in I}\phi \left(e_i\right)\overline{\phi \left(e_i\right)}=\sum _{i\in I}{\left|\phi \left(e_i\right)\right|}^2=\Vert \phi {\Vert }^2.}$$

If the orthonormal basis${\displaystyle {\left\{e_i\right\}}_{i\in I}={\left\{e_i\right\}}_{i=1}^{\mathrm{\infty }}}$ is a sequence then this becomes$${\displaystyle f_{\phi }=\overline{\phi \left(e_1\right)}{e}_1+\overline{\phi \left(e_2\right)}{e}_2+\cdots }$$ and if${\displaystyle y\in H}$ is written as${\displaystyle y=\sum _{i\in I}{a}_i{e}_i=a_1{e}_1+a_2{e}_2+\cdots }$ then$${\displaystyle \phi (y)=\phi \left(e_1\right)a_1+\phi \left(e_2\right)a_2+\cdots =⟨f_{\phi }|y⟩.}$$

Consider the special case of${\displaystyle H={\mathbb{C}}^n}$ (where${\displaystyle n>0}$ is aninteger) with the standard inner product$${\displaystyle ⟨z\mid w⟩:={\overline{\overrightarrow{z}}}^{\mathrm{T}}\overrightarrow{w}\text{ for all }w,z\in H}$$ where${\displaystyle w\text{ and }z}$ are represented ascolumn matrices${\displaystyle \overrightarrow{w}:=\left[\begin{array}{c}{w}_1\\ ⋮\\ w_n\end{array}\right]}$ and${\displaystyle \overrightarrow{z}:=\left[\begin{array}{c}{z}_1\\ ⋮\\ z_n\end{array}\right]}$ with respect to the standard orthonormal basis${\displaystyle e_1,\dots ,e_n}$ on${\displaystyle H}$ (here,${\displaystyle e_i}$ is${\displaystyle 1}$ at its${\displaystyle i}$^th coordinate and${\displaystyle 0}$ everywhere else; as usual,${\displaystyle H^{\ast }}$ will now be associated with thedual basis) and where${\displaystyle {\overline{\overrightarrow{z}}}^{\mathrm{T}}:=\left[\overline{z_1},\dots ,\overline{z_n}\right]}$ denotes theconjugate transpose of${\displaystyle \overrightarrow{z}.}$ Let${\displaystyle \phi \in H^{\ast }}$ be any linear functional and let${\displaystyle {\phi }_1,\dots ,{\phi }_n\in \mathbb{C}}$ be the unique scalars such that$${\displaystyle \phi \left(w_1,\dots ,w_n\right)={\phi }_1{w}_1+\cdots +{\phi }_n{w}_n\text{ for all }w:=\left(w_1,\dots ,w_n\right)\in H,}$$ where it can be shown that${\displaystyle {\phi }_i=\phi \left(e_i\right)}$ for all${\displaystyle i=1,\dots ,n.}$ Then the Riesz representation of${\displaystyle \phi }$ is the vector$${\displaystyle f_{\phi }:=\overline{{\phi }_1}{e}_1+\cdots +\overline{{\phi }_n}{e}_n=\left(\overline{{\phi }_1},\dots ,\overline{{\phi }_n}\right)\in H.}$$To see why, identify every vector${\displaystyle w=\left(w_1,\dots ,w_n\right)}$ in${\displaystyle H}$ with the column matrix${\displaystyle \overrightarrow{w}:=\left[\begin{array}{c}{w}_1\\ ⋮\\ w_n\end{array}\right]}$ so that${\displaystyle f_{\phi }}$ is identified with${\displaystyle \overrightarrow{f_{\phi }}:=\left[\begin{array}{c}\overline{{\phi }_1}\\ ⋮\\ \overline{{\phi }_n}\end{array}\right]=\left[\begin{array}{c}\overline{\phi \left(e_1\right)}\\ ⋮\\ \overline{\phi \left(e_n\right)}\end{array}\right].}$ As usual, also identify the linear functional${\displaystyle \phi }$ with itstransformation matrix, which is therow matrix${\displaystyle \overrightarrow{\phi }:=\left[{\phi }_1,\dots ,{\phi }_n\right]}$ so that${\displaystyle \overrightarrow{f_{\phi }}:={\overline{\overrightarrow{\phi }}}^{\mathrm{T}}}$ and the function${\displaystyle \phi }$ is the assignment${\displaystyle \overrightarrow{w}\mapsto \overrightarrow{\phi }\overrightarrow{w},}$ where the right hand side ismatrix multiplication. Then for all${\displaystyle w=\left(w_1,\dots ,w_n\right)\in H,}$$${\displaystyle \phi (w)={\phi }_1{w}_1+\cdots +{\phi }_n{w}_n=\left[{\phi }_1,\dots ,{\phi }_n\right]\left[\begin{array}{c}{w}_1\\ ⋮\\ w_n\end{array}\right]={\overline{\left[\begin{array}{c}\overline{{\phi }_1}\\ ⋮\\ \overline{{\phi }_n}\end{array}\right]}}^{\mathrm{T}}\overrightarrow{w}={\overline{\overrightarrow{f_{\phi }}}}^{\mathrm{T}}\overrightarrow{w}=⟨f_{\phi }\mid w⟩,}$$which shows that${\displaystyle f_{\phi }}$ satisfies the defining condition of the Riesz representation of${\displaystyle \phi .}$ The bijective antilinear isometry${\displaystyle \mathrm{\Phi }:H\to H^{\ast }}$ defined in the corollary to the Riesz representation theorem is the assignment that sends${\displaystyle z=\left(z_1,\dots ,z_n\right)\in H}$ to the linear functional${\displaystyle \mathrm{\Phi }(z)\in H^{\ast }}$ on${\displaystyle H}$ defined by$${\displaystyle w=\left(w_1,\dots ,w_n\right)\mapsto ⟨z\mid w⟩=\overline{z_1}{w}_1+\cdots +\overline{z_n}{w}_n,}$$ where under the identification of vectors in${\displaystyle H}$ with column matrices and vector in${\displaystyle H^{\ast }}$ with row matrices,${\displaystyle \mathrm{\Phi }}$ is just the assignment$${\displaystyle \overrightarrow{z}=\left[\begin{array}{c}{z}_1\\ ⋮\\ z_n\end{array}\right]\mapsto {\overline{\overrightarrow{z}}}^{\mathrm{T}}=\left[\overline{z_1},\dots ,\overline{z_n}\right].}$$ As described in the corollary,${\displaystyle \mathrm{\Phi }}$'s inverse${\displaystyle {\mathrm{\Phi }}^{-1}:H^{\ast }\to H}$ is the antilinear isometry${\displaystyle \phi \mapsto f_{\phi },}$ which was just shown above to be:$${\displaystyle \phi \mapsto f_{\phi }:=\left(\overline{\phi \left(e_1\right)},\dots ,\overline{\phi \left(e_n\right)}\right);}$$ where in terms of matrices,${\displaystyle {\mathrm{\Phi }}^{-1}}$ is the assignment$${\displaystyle \overrightarrow{\phi }=\left[{\phi }_1,\dots ,{\phi }_n\right]\mapsto {\overline{\overrightarrow{\phi }}}^{\mathrm{T}}=\left[\begin{array}{c}\overline{{\phi }_1}\\ ⋮\\ \overline{{\phi }_n}\end{array}\right].}$$ Thus in terms of matrices, each of${\displaystyle \mathrm{\Phi }:H\to H^{\ast }}$ and${\displaystyle {\mathrm{\Phi }}^{-1}:H^{\ast }\to H}$ is just the operation ofconjugate transposition${\displaystyle \overrightarrow{v}\mapsto {\overline{\overrightarrow{v}}}^{\mathrm{T}}}$ (although between different spaces of matrices: if${\displaystyle H}$ is identified with the space of all column (respectively, row) matrices then${\displaystyle H^{\ast }}$ is identified with the space of all row (respectively, column matrices).