Riesz representation theorem

The Riesz representation theorem, sometimes called the Riesz-Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism.

Let \(H\) be a Hilbert space over a field \(\mathbb{F},\) where \(\mathbb{F}\) is either the real numbers \(\R\) or the complex numbers \(\Complex.\) If \(\mathbb{F} = \Complex\) (resp. if \(\mathbb{F} = \R\)) then \(H\) is called a complex Hilbert space (resp. a real Hilbert space). Every real Hilbert space can be extended to be a dense subset of a unique (up to bijective isometry) complex Hilbert space, called its complexification, which is why Hilbert spaces are often automatically assumed to be complex. Real and complex Hilbert spaces have in common many, but by no means all, properties and results/theorems.

This article is intended for both mathematicians and physicists and will describe the theorem for both. In both mathematics and physics, if a Hilbert space is assumed to be real (that is, if \(\mathbb{F} = \R\)) 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) \(f : H \to Y\) is a map between vector spaces that is additive: \[f(x + y) = f(x) + f(y) \quad \text{ for all } x, y \in H,\] and antilinear (also called conjugate-linear or conjugate-homogeneous): \[f(c x) = \overline{c} f(x) \quad \text{ for all } x \in H \text{ and all scalar } c \in \mathbb{F},\] where \(\overline{c}\) is the conjugate of the complex number \(c = a + b i\), given by \(\overline{c} = a - b i\).

In contrast, a map \(f : H \to Y\) is linear if it is additive and homogeneous: \[f(c x) = c f(x) \quad \text{ for all } x \in H \quad \text{ and all scalars } c \in \mathbb{F}.\]

Every constant \(0\) map is always both linear and antilinear. If \(\mathbb{F} = \R\) then the definitions of linear maps and antilinear maps are completely identical. A linear map from a Hilbert space into a Banach space (or more generally, from any Banach space into any topological vector space) is continuous if and only if it is bounded; the same is true of antilinear maps. The inverse of any antilinear (resp. linear) bijection is again an antilinear (resp. linear) bijection. The composition of two antilinear maps is a linear map.

Continuous dual and anti-dual spaces

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

One-to-one correspondence between linear and antilinear functionals

Given any functional \(f ~:~ H \to \mathbb{F},\) the conjugate of \(f\) is the functional \[\begin{alignedat}{4} \overline{f} : \,& H && \to \,&& \mathbb{F} \\ & h && \mapsto\,&& \overline{f(h)}. \\ \end{alignedat}\]

all functionals (resp. all linear functionals, all continuous linear functionals \(H^*\)) on \(H,\)

all functionals (resp. all antilinear functionals, all continuous antilinear functionals \(\overline{H}^*\)) on \(H.\)

Condensed: the full section is in Wikipedia.

The Hilbert space \(H\) has an associated inner product \(H \times H \to \mathbb{F}\) valued in \(H\)'s underlying scalar field \(\mathbb{F}\) that is linear in one coordinate and antilinear in the other (as specified below). If \(H\) is a complex Hilbert space (\(\mathbb{F} = \Complex\)), 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 (\(\mathbb{F} = \R\)), 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 \(H\) is often denoted by \(\left\langle \cdot\,, \cdot \right\rangle\) or \(\left\langle \cdot\,, \cdot \right\rangle_H\) while in physics, the bra, ket notation \(\left\langle \cdot \mid \cdot \right\rangle\) or \(\left\langle \cdot \mid \cdot \right\rangle_H\) is typically used. In this article, these two notations will be related by the equality:

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

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

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

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

defines a canonical norm on \(H\) that makes \(H\) into a normed space. As with all normed spaces, the (continuous) dual space \(H^*\) carries a canonical norm, called the dual norm, that is defined by \[\|f\|_{H^*} ~:=~ \sup_{\|x\| \leq 1, x \in H} |f(x)| \quad \text{ for every } f \in H^*.\]

The canonical norm on the (continuous) anti-dual space \(\overline{H}^*,\) denoted by \(\|f\|_{\overline{H}^*},\) is defined by using this same equation: \[\|f\|_{\overline{H}^*} ~:=~ \sup_{\|x\| \leq 1, x \in H} |f(x)| \quad \text{ for every } f \in \overline{H}^*.\]

This canonical norm on \(H^*\) satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on \(H^*,\) which this article will denote by the notations \[\left\langle f, g \right\rangle_{H^*} := \left\langle g \mid f \right\rangle_{H^*},\] where this inner product turns \(H^*\) into a Hilbert space. There are now two ways of defining a norm on \(H^*:\) the norm induced by this inner product (that is, the norm defined by \(f \mapsto \sqrt{\left\langle f, f \right\rangle_{H^*}}\)) and the usual dual norm (defined as the supremum over the closed unit ball). These norms are the same; explicitly, this means that the following holds for every \(f \in H^*:\) \[\sup_{\|x\| \leq 1, x \in H} |f(x)| = \|f\|_{H^*} ~=~ \sqrt{\langle f, f \rangle_{H^*}} ~=~ \sqrt{\langle f \mid f \rangle_{H^*}}.\]

As will be described later, the Riesz representation theorem can be used to give an equivalent definition of the canonical norm and the canonical inner product on \(H^*.\)

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

Canonical isometry between the dual and antidual

Condensed: the full section is in Wikipedia.

Two vectors \(x\) and \(y\) are orthogonal if \(\langle x, y \rangle = 0,\) which happens if and only if \(\|y\| \leq \|y + s x\|\) for all scalars \(s.\) The orthogonal complement of a subset \(X \subseteq H\) is \[X^{\bot} := \{ \,y \in H : \langle y, x \rangle = 0 \text{ for all } x \in X\, \},\] which is always a closed vector subspace of \(H.\) The Hilbert projection theorem guarantees that for any nonempty closed convex subset \(C\) of a Hilbert space there exists a unique vector \(m \in C\) such that \(\|m\| = \inf_{c \in C} \|c\|;\) that is, \(m \in C\) is the (unique) global minimum point of the function \(C \to [0, \infty)\) defined by \(c \mapsto \|c\|.\)

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

Importantly for complex Hilbert spaces, \(f_{\varphi}\) is always located in the antilinear coordinate of the inner product.

Furthermore, the length of the representation vector is equal to the norm of the functional: \[\left\|f_\varphi\right\|_H = \|\varphi\|_{H^*},\] and \(f_{\varphi}\) is the unique vector \(f_{\varphi} \in \left(\ker \varphi\right)^{\bot}\) with \(\varphi\left(f_{\varphi}\right) = \|\varphi\|^2.\) It is also the unique element of minimum norm in \(C := \varphi^{-1}\left(\|\varphi\|^2\right)\); that is to say, \(f_{\varphi}\) is the unique element of \(C\) satisfying \(\left\|f_{\varphi}\right\| = \inf_{c \in C} \|c\|.\) Moreover, any non-zero \(q \in (\ker \varphi)^{\bot}\) can be written as \(q = \left(\|q\|^2 /\, \overline{\varphi(q)}\right)\ f_{\varphi}.\)

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

Condensed: the full section is in Wikipedia.

If \(\varphi \in H^*\) then \[\varphi \left(f_{\varphi}\right) = \left\langle f_{\varphi}, f_{\varphi} \right\rangle = \left\|f_{\varphi}\right\|^2 = \|\varphi\|^2.\] So in particular, \(\varphi \left(f_{\varphi}\right) \geq 0\) is always real and furthermore, \(\varphi \left(f_{\varphi}\right) = 0\) if and only if \(f_{\varphi} = 0\) if and only if \(\varphi = 0.\)

Linear functionals as affine hyperplanes

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

Using the notation from the theorem above, several ways of constructing \(f_{\varphi}\) from \(\varphi \in H^*\) are now described. If \(\varphi = 0\) then \(f_{\varphi} := 0\); in other words, \[f_0 = 0.\]

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

Orthogonal complement of kernel

If \(\varphi \neq 0\) then for any \(0 \neq u \in (\ker\varphi)^{\bot},\) \[f_{\varphi} := \frac{\overline{\varphi(u)} u}{\|u\|^2}.\]

If \(u \in (\ker\varphi)^{\bot}\) is a unit vector (meaning \(\|u\| = 1\)) then \[f_{\varphi} := \overline{\varphi(u)} u\] (this is true even if \(\varphi = 0\) because in this case \(f_{\varphi} = \overline{\varphi(u)} u = \overline{0} u = 0\)). If \(u\) is a unit vector satisfying the above condition then the same is true of \(-u,\) which is also a unit vector in \((\ker\varphi)^{\bot}.\) However, \(\overline{\varphi(-u)} (-u) = \overline{\varphi(u)} u = f_\varphi\) so both these vectors result in the same \(f_{\varphi}.\)

Orthogonal projection onto kernel

If \(x \in H\) is such that \(\varphi(x) \neq 0\) and if \(x_K\) is the orthogonal projection of \(x\) onto \(\ker\varphi\) then \[f_{\varphi} = \frac{\|\varphi\|^2}{\varphi(x)} \left(x - x_K\right).\]

Condensed: the full section is in Wikipedia.

Assume that \(H\) is a complex Hilbert space with inner product \(\langle \,\cdot\mid\cdot\, \rangle.\) When the Hilbert space \(H\) is reinterpreted as a real Hilbert space then it will be denoted by \(H_{\R},\) where the (real) inner-product on \(H_{\R}\) is the real part of \(H\)'s inner product; that is: \[\langle x, y \rangle_{\R} := \operatorname{re} \langle x, y \rangle.\]

The norm on \(H_{\R}\) induced by \(\langle \,\cdot\,, \,\cdot\, \rangle_{\R}\) is equal to the original norm on \(H\) and the continuous dual space of \(H_{\R}\) is the set of all real-valued bounded \(\R\)-linear functionals on \(H_{\R}\) (see the article about the polarization identity for additional details about this relationship). Let \(\psi_{\R} := \operatorname{re} \psi\) and \(\psi_{i} := \operatorname{im} \psi\) denote the real and imaginary parts of a linear functional \(\psi,\) so that \(\psi = \operatorname{re} \psi + i \operatorname{im} \psi = \psi_{\R} + i \psi_{i}.\) The formula expressing a linear functional in terms of its real part is \[\psi(h) = \psi_{\R}(h) - i \psi_{\R} (i h) \quad \text{ for } h \in H,\] where \(\psi_{i}(h) = - i \psi_{\R} (i h)\) for all \(h \in H.\) It follows that \(\ker\psi_{\R} = \psi^{-1}(i \R),\) and that \(\psi = 0\) if and only if \(\psi_{\R} = 0.\) It can also be shown that \(\|\psi\| = \left\|\psi_{\R}\right\| = \left\|\psi_i\right\|\) where \(\left\|\psi_{\R}\right\| := \sup_{\|h\| \leq 1} \left|\psi_{\R}(h)\right|\) and \(\left\|\psi_i\right\| := \sup_{\|h\| \leq 1} \left|\psi_i(h)\right|\) are the usual operator norms. In particular, a linear functional \(\psi\) is bounded if and only if its real part \(\psi_{\R}\) is bounded.

Representing a functional and its real part

The Riesz representation of a continuous linear function \(\varphi\) on a complex Hilbert space is equal to the Riesz representation of its real part \(\operatorname{re} \varphi\) on its associated real Hilbert space.

Furthermore, if \(\varphi \neq 0\) then \(f_{\varphi}\) is perpendicular to \(\ker\varphi_{\R}\) with respect to \(\langle \cdot, \cdot \rangle_{\R}\) where the kernel of \(\varphi\) is be a proper subspace of the kernel of its real part \(\varphi_{\R}.\) Assume now that \(\varphi \neq 0.\) Then \(f_{\varphi} \not\in \ker\varphi_{\R}\) because \(\varphi_{\R}\left(f_{\varphi}\right) = \varphi\left(f_{\varphi}\right) = \|\varphi\|^2 \neq 0\) and \(\ker\varphi\) is a proper subset of \(\ker\varphi_{\R}.\) The vector subspace \(\ker \varphi\) has real codimension \(1\) in \(\ker\varphi_{\R},\) while \(\ker\varphi_{\R}\) has real codimension \(1\) in \(H_{\R},\) and \(\left\langle f_{\varphi}, \ker\varphi_{\R} \right\rangle_{\R} = 0.\) That is, \(f_{\varphi}\) is perpendicular to \(\ker\varphi_{\R}\) with respect to \(\langle \cdot, \cdot \rangle_{\R}.\)

Condensed: the full section is in Wikipedia.

Induced linear map into anti-dual

The map defined by placing \(y\) into the linear coordinate of the inner product and letting the variable \(h \in H\) vary over the antilinear coordinate results in an antilinear functional: \[\langle \,\cdot \mid y\, \rangle = \langle \,y, \cdot\, \rangle : H \to \mathbb{F} \quad \text{ defined by } \quad h \mapsto \langle \,h \mid y\, \rangle = \langle \,y, h\, \rangle.\]

This map is an element of \(\overline{H}^*,\) which is the continuous anti-dual space of \(H.\) The canonical map from \(H\) into its anti-dual \(\overline{H}^*\) is the linear operator \[\begin{alignedat}{4} \operatorname{In}_H^{\overline{H}^*} :\;&& H &&\;\to \;& \overline{H}^* \\[0.3ex] && y &&\;\mapsto\;& \langle \,\cdot \mid y\, \rangle = \langle \,y, \cdot\, \rangle \\[0.3ex] \end{alignedat}\] which is also an injective isometry. 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 \(H\) can be written (uniquely) in this form.

If \(\operatorname{Cong} : H^* \to \overline{H}^*\) is the canonical antilinear bijective isometry \(f \mapsto \overline{f}\) that was defined above, then the following equality holds: \[\operatorname{Cong} ~\circ~ \operatorname{In}_H^{H^*} ~=~ \operatorname{In}_H^{\overline{H}^*}.\]

Let \(\left(H, \langle\cdot, \cdot \rangle_H\right)\) be a Hilbert space and as before, let \(\langle y\, | \,x \rangle_H := \langle x, y \rangle_H.\) Let \[\begin{alignedat}{4} \Phi :\;&& H &&\;\to \;& H^* \\[0.3ex] && g &&\;\mapsto\;& \left\langle \,g\mid \cdot\, \right\rangle_H = \left\langle \,\cdot, g\, \right\rangle_H \\ \end{alignedat}\] which is a bijective antilinear isometry that satisfies \[(\Phi h) g = \langle h\mid g \rangle_H = \langle g, h \rangle_H \quad \text{ for all } g, h \in H.\]

Bras

Given a vector \(h \in H,\) let \(\langle h\, |\) denote the continuous linear functional \(\Phi h\); that is, \[\langle h\, | ~:=~ \Phi h\] so that this functional \(\langle h\, |\) is defined by \(g \mapsto \left\langle \,h\mid g\, \right\rangle_H.\) This map was denoted by \(\left\langle h \mid \cdot\, \right\rangle\) earlier in this article.

The assignment \(h \mapsto \langle h |\) is just the isometric antilinear isomorphism \(\Phi ~:~ H \to H^*,\) which is why \(~\langle c g + h\, | ~=~ \overline{c} \langle g\mid ~+~ \langle h\, |~\) holds for all \(g, h \in H\) and all scalars \(c.\) The result of plugging some given \(g \in H\) into the functional \(\langle h\, |\) is the scalar \(\langle h\, | \,g \rangle_H = \langle g, h \rangle_H,\) which may be denoted by \(\langle h \mid g \rangle.\)

Bra of a linear functional

Given a continuous linear functional \(\psi \in H^*,\) let \(\langle \psi\mid\) denote the vector \(\Phi^{-1} \psi \in H\); that is, \[\langle \psi\mid ~:=~ \Phi^{-1} \psi.\]

The assignment \(\psi \mapsto \langle \psi\mid\) is just the isometric antilinear isomorphism \(\Phi^{-1} ~:~ H^* \to H,\) which is why \(~\langle c \psi + \phi\mid ~=~ \overline{c} \langle \psi\mid ~+~ \langle \phi\mid~\) holds for all \(\phi, \psi \in H^*\) and all scalars \(c.\)

Condensed: the full section is in Wikipedia.

Let \(A : H \to Z\) be a continuous linear operator between Hilbert spaces \(\left(H, \langle \cdot, \cdot \rangle_H\right)\) and \(\left(Z, \langle \cdot, \cdot \rangle_Z \right).\) As before, let \(\langle y \mid x \rangle_H := \langle x, y \rangle_H\) and \(\langle y \mid x \rangle_Z := \langle x, y \rangle_Z.\)

Denote by \[\begin{alignedat}{4} \Phi_H :\;&& H &&\;\to \;& H^* \\[0.3ex] && g &&\;\mapsto\;& \langle \,g \mid \cdot\, \rangle_H \\ \end{alignedat} \quad \text{ and } \quad \begin{alignedat}{4} \Phi_Z :\;&& Z &&\;\to \;& Z^* \\[0.3ex] && y &&\;\mapsto\;& \langle \,y \mid \cdot\, \rangle_Z \\ \end{alignedat}\] the usual bijective antilinear isometries that satisfy: \[\left(\Phi_H g\right) h = \langle g\mid h \rangle_H \quad \text{ for all } g, h \in H \qquad \text{ and } \qquad \left(\Phi_Z y\right) z = \langle y \mid z \rangle_Z \quad \text{ for all } y, z \in Z.\]

For every \(z \in Z,\) the scalar-valued map \(\langle z\mid A (\cdot) \rangle_Z\) on \(H\) defined by \[h \mapsto \langle z\mid A h \rangle_Z = \langle A h, z \rangle_Z\]

is a continuous linear functional on \(H\) and so by the Riesz representation theorem, there exists a unique vector in \(H,\) denoted by \(A^* z,\) such that \(\langle z \mid A (\cdot) \rangle_Z = \left\langle A^* z \mid \cdot\, \right\rangle_H,\) or equivalently, such that \[\langle z \mid A h \rangle_Z = \left\langle A^* z \mid h \right\rangle_H \quad \text{ for all } h \in H.\]

The assignment \(z \mapsto A^* z\) thus induces a function \(A^* : Z \to H\) called the adjoint of \(A : H \to Z\) whose defining condition is \[\langle z \mid A h \rangle_Z = \left\langle A^* z\mid h \right\rangle_H \quad \text{ for all } h \in H \text{ and all } z \in Z.\] The adjoint \(A^* : Z \to H\) is necessarily a continuous (equivalently, a bounded) linear operator.

If \(H\) is finite dimensional with the standard inner product and if \(M\) is the transformation matrix of \(A\) with respect to the standard orthonormal basis then \(M\)'s conjugate transpose \(\overline{M^{\operatorname{T}}}\) is the transformation matrix of the adjoint \(A^*.\)