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.

By definition, an antilinear map (also called a conjugate-linear map) ${\displaystyle f:H\to Y}$  is a map between vector spaces that is additive: $${\displaystyle f(x+y)=f(x)+f(y)\quad {\text{ for all }}x,y\in H,}$$  and antilinear (also called conjugate-linear or conjugate-homogeneous): $${\displaystyle f(cx)={\overline {c}}f(x)\quad {\text{ for all }}x\in H{\text{ and all scalar }}c\in \mathbb {F} ,}$$  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 ${\displaystyle f:H\to Y}$  is linear if it is additive and homogeneous: $${\displaystyle f(cx)=cf(x)\quad {\text{ for all }}x\in H\quad {\text{ and all scalars }}c\in \mathbb {F} .}$$ 

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 ${\displaystyle H\to \mathbb {F} }$  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 ${\displaystyle f~:~H\to \mathbb {F} ,}$  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 ${\displaystyle f\mapsto {\overline {f}}}$  reduces down to the identity map.

The assignment ${\displaystyle f\mapsto {\overline {f}}}$  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 ${\displaystyle H.}$ 

The Hilbert space ${\displaystyle H}$  has an associated inner 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 a symmetric map that is linear in each coordinate (bilinear), so there can be no such confusion.

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

$${\displaystyle \left\langle x,y\right\rangle :=\left\langle y\mid x\right\rangle \quad {\text{ for all }}x,y\in H.}$$ These have the following properties:

1. The map ${\displaystyle \left\langle \cdot \,,\cdot \right\rangle }$  is linear in its first coordinate; equivalently, the map ${\displaystyle \left\langle \cdot \mid \cdot \right\rangle }$  is linear in its second coordinate. That is, for fixed ${\displaystyle y\in H,}$  the map ${\displaystyle \left\langle \,y\mid \cdot \,\right\rangle =\left\langle \,\cdot \,,y\,\right\rangle :H\to \mathbb {F} }$  with ${\textstyle h\mapsto \left\langle \,y\mid h\,\right\rangle =\left\langle \,h,y\,\right\rangle }$  is a linear functional on ${\displaystyle H.}$  This linear functional is continuous, so ${\displaystyle \left\langle \,y\mid \cdot \,\right\rangle =\left\langle \,\cdot ,y\,\right\rangle \in H^{*}.}$ 
2. The map ${\displaystyle \left\langle \cdot \,,\cdot \right\rangle }$  is antilinear in its second coordinate; equivalently, the map ${\displaystyle \left\langle \cdot \mid \cdot \right\rangle }$  is antilinear in its first coordinate. That is, for fixed ${\displaystyle y\in H,}$  the map ${\displaystyle \left\langle \,\cdot \mid y\,\right\rangle =\left\langle \,y,\cdot \,\right\rangle :H\to \mathbb {F} }$  with ${\textstyle h\mapsto \left\langle \,h\mid y\,\right\rangle =\left\langle \,y,h\,\right\rangle }$  is an antilinear functional on ${\displaystyle H.}$  This antilinear functional is continuous, so ${\displaystyle \left\langle \,\cdot \mid y\,\right\rangle =\left\langle \,y,\cdot \,\right\rangle \in {\overline {H}}^{*}.}$ 

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

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

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

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

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 $${\displaystyle \left\langle f,g\right\rangle _{H^{*}}:=\left\langle g\mid f\right\rangle _{H^{*}},}$$  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 ${\displaystyle f\mapsto {\sqrt {\left\langle f,f\right\rangle _{H^{*}}}}}$ ) and the usual dual norm (defined as the supremum over the closed unit ball). These norms are the same; explicitly, this means that the following holds for every ${\displaystyle f\in H^{*}:}$  $${\displaystyle \sup _{\|x\|\leq 1,x\in H}|f(x)|=\|f\|_{H^{*}}~=~{\sqrt {\langle f,f\rangle _{H^{*}}}}~=~{\sqrt {\langle f\mid f\rangle _{H^{*}}}}.}$$ 

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 ${\displaystyle f\in H^{*}}$  and every ${\displaystyle g\in {\overline {H}}^{*}.}$  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 $${\displaystyle \langle \,{\overline {f}}\,|\,{\overline {g}}\,\rangle _{{\overline {H}}^{*}}={\overline {\langle \,f\,|\,g\,\rangle _{H^{*}}}}=\langle \,g\,|\,f\,\rangle _{H^{*}}\qquad {\text{ for all }}f,g\in H^{*}}$$  and $${\displaystyle \langle \,{\overline {f}}\,|\,{\overline {g}}\,\rangle _{H^{*}}={\overline {\langle \,f\,|\,g\,\rangle _{{\overline {H}}^{*}}}}=\langle \,g\,|\,f\,\rangle _{{\overline {H}}^{*}}\qquad {\text{ for all }}f,g\in {\overline {H}}^{*}.}$$ 

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.

Two vectors ${\displaystyle x}$  and ${\displaystyle y}$  are orthogonal if ${\displaystyle \langle x,y\rangle =0,}$  which happens if and only if ${\displaystyle \|y\|\leq \|y+sx\|}$  for all scalars ${\displaystyle s.}$ ^[2] The orthogonal complement of a subset ${\displaystyle X\subseteq H}$  is $${\displaystyle X^{\bot }:=\{\,y\in H:\langle y,x\rangle =0{\text{ for all }}x\in X\,\},}$$  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 ${\displaystyle m\in C}$  such that ${\displaystyle \|m\|=\inf _{c\in C}\|c\|;}$  that is, ${\displaystyle m\in C}$  is the (unique) global minimum point of the function ${\displaystyle C\to [0,\infty )}$  defined by ${\displaystyle c\mapsto \|c\|.}$ 

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

If ${\displaystyle \varphi \in H^{*}}$  then $${\displaystyle \varphi \left(f_{\varphi }\right)=\left\langle f_{\varphi },f_{\varphi }\right\rangle =\left\|f_{\varphi }\right\|^{2}=\|\varphi \|^{2}.}$$  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 ${\displaystyle \|\varphi \|=\left\|f_{\varphi }\right\|=\inf _{c\in C}\|c\|}$  implies ${\displaystyle \|\varphi \|=\inf _{a\in A}\|\varphi \|^{2}\|a\|}$  and thus ${\displaystyle \|\varphi \|={\frac {1}{\inf _{a\in A}\|a\|}}.}$  This can also be seen by applying the Hilbert projection theorem to ${\displaystyle A}$  and concluding that the global minimum point of the map ${\displaystyle A\to [0,\infty )}$  defined by ${\displaystyle a\mapsto \|a\|}$  is ${\displaystyle {\frac {f_{\varphi }}{\|\varphi \|^{2}}}\in A.}$  The formulas $${\displaystyle {\frac {1}{\inf _{a\in A}\|a\|}}=\sup _{a\in A}{\frac {1}{\|a\|}}}$$  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 $${\displaystyle \|\varphi \|={\frac {1}{\inf _{a\in \varphi ^{-1}(1)}\|a\|}}}$$  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 in 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 ${\displaystyle \|\varphi \|=\sup _{a\in \varphi ^{-1}(1)}{\frac {1}{\|a\|}}}$  will also hold when ${\displaystyle \varphi =0}$  (although the atypical equality ${\displaystyle \sup \varnothing =0}$  is usually unexpected and so risks causing confusion).

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 ${\displaystyle 0\neq u\in (\ker \varphi )^{\bot },}$  $${\displaystyle f_{\varphi }:={\frac {{\overline {\varphi (u)}}u}{\|u\|^{2}}}.}$$ 

If ${\displaystyle u\in (\ker \varphi )^{\bot }}$  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 ${\displaystyle x\in H}$  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 ${\displaystyle \left\{e_{i}\right\}_{i\in I}}$  of ${\displaystyle H}$  and a continuous linear functional ${\displaystyle \varphi \in H^{*},}$  the vector ${\displaystyle f_{\varphi }\in H}$  can be constructed uniquely by $${\displaystyle f_{\varphi }=\sum _{i\in I}{\overline {\varphi \left(e_{i}\right)}}e_{i}}$$  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 ${\displaystyle y\in H}$  is written as ${\displaystyle y=\sum _{i\in I}a_{i}e_{i}}$  then $${\displaystyle \varphi (y)=\sum _{i\in I}\varphi \left(e_{i}\right)a_{i}=\langle f_{\varphi }|y\rangle }$$  and $${\displaystyle \left\|f_{\varphi }\right\|^{2}=\varphi \left(f_{\varphi }\right)=\sum _{i\in I}\varphi \left(e_{i}\right){\overline {\varphi \left(e_{i}\right)}}=\sum _{i\in I}\left|\varphi \left(e_{i}\right)\right|^{2}=\|\varphi \|^{2}.}$$ 

If the orthonormal basis ${\displaystyle \left\{e_{i}\right\}_{i\in I}=\left\{e_{i}\right\}_{i=1}^{\infty }}$  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 ${\displaystyle y\in H}$  is written as ${\displaystyle y=\sum _{i\in I}a_{i}e_{i}=a_{1}e_{1}+a_{2}e_{2}+\cdots }$  then $${\displaystyle \varphi (y)=\varphi \left(e_{1}\right)a_{1}+\varphi \left(e_{2}\right)a_{2}+\cdots =\langle f_{\varphi }|y\rangle .}$$ 

Consider the special case of ${\displaystyle H=\mathbb {C} ^{n}}$  (where ${\displaystyle n>0}$  is an integer) with the standard inner product $${\displaystyle \langle z\mid w\rangle :={\overline {\,{\vec {z}}\,\,}}^{\operatorname {T} }{\vec {w}}\qquad {\text{ for all }}\;w,z\in H}$$  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 ${\displaystyle w=\left(w_{1},\ldots ,w_{n}\right)\in H,}$  $${\displaystyle \varphi (w)=\varphi _{1}w_{1}+\cdots +\varphi _{n}w_{n}=\left[\varphi _{1},\ldots ,\varphi _{n}\right]{\begin{bmatrix}w_{1}\\\vdots \\w_{n}\end{bmatrix}}={\overline {\begin{bmatrix}{\overline {\varphi _{1}}}\\\vdots \\{\overline {\varphi _{n}}}\end{bmatrix}}}^{\operatorname {T} }{\vec {w}}={\overline {\,{\vec {f_{\varphi }}}\,\,}}^{\operatorname {T} }{\vec {w}}=\left\langle \,\,f_{\varphi }\,\mid \,w\,\right\rangle ,}$$  which shows that ${\displaystyle f_{\varphi }}$  satisfies the defining condition of the Riesz representation of ${\displaystyle \varphi .}$  The bijective antilinear isometry ${\displaystyle \Phi :H\to H^{*}}$  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 ${\displaystyle \Phi :H\to H^{*}}$  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).