Riesz Representation in Finite Dimensions

Statement

Let \( V \) be a finite-dimensional inner-product space over the field \( \mathbb{F} \) (either \( \mathbb{R} \) or \( \mathbb{C} \)) with inner product \( \langle \cdot , \cdot \rangle \), taken linear in its first argument and conjugate-linear in its second. Then for every linear functional \( \varphi : V \to \mathbb{F} \) there exists a unique vector \( \mathbf{r}_\varphi \in V \), the Riesz representative, such that \[ \varphi(\mathbf{x}) = \langle \mathbf{x}, \mathbf{r}_\varphi \rangle \qquad \text{for all } \mathbf{x} \in V. \] The assignment \( \varphi \mapsto \mathbf{r}_\varphi \) is a conjugate-linear bijection from the dual space \( V^{*} \) onto \( V \).

Why it matters

The theorem identifies a space with its own dual in a way that requires no arbitrary choice of basis: once an inner product is fixed, every "measurement" (linear functional) is realised as taking the inner product with one physical vector. This is the finite-dimensional root of the position/momentum bras and kets of quantum mechanics, the gradient-as-a-vector identification in mechanics, and the very definition of the adjoint operator.

Because the correspondence is canonical, it turns statements about functionals — often awkward, living in an abstract dual — into statements about vectors, which are concrete and visualisable. Symmetry threads run straight through: the map is an isometry of the natural norms, and it is the linear-algebra shadow of the far deeper Hilbert-space Riesz theorem that underpins the whole bra-ket formalism.

Assumptions

Derivation

Result

Limiting cases

• \( \varphi = 0 \): the formula gives \( \mathbf{r}_\varphi = \mathbf{0} \), consistent with \( \langle \mathbf{x},\mathbf{0}\rangle = 0 \) for all \( \mathbf{x} \).
• \( \mathbb{F} = \mathbb{R} \): all conjugations drop out, the map \( \varphi\mapsto\mathbf{r}_\varphi \) is genuinely linear (not merely conjugate-linear), and \( \mathbf{r}_\varphi \) is the gradient covector raised to a vector.
• \( n = 1 \): \( V \cong \mathbb{F} \), \( \varphi(x) = cx \), and \( \mathbf{r}_\varphi = \bar c \); the theorem reduces to a \( 1\times 1 \) matrix being its own conjugate-transpose action.
• Standard \( \mathbb{R}^n \) with the dot product: \( \varphi(\mathbf{x}) = \mathbf{a}^{\top}\mathbf{x} \) has \( \mathbf{r}_\varphi = \mathbf{a} \); the row vector and column vector are identified.

Breaks when

Failure modes

• Dropping the conjugate. Writing \( \mathbf{r}_\varphi = \sum \varphi(\mathbf{e}_i)\,\mathbf{e}_i \) over \( \mathbb{C} \). This satisfies \( \varphi(\mathbf{x}) = \overline{\langle \mathbf{r}_\varphi,\mathbf{x}\rangle} \), not \( \langle\mathbf{x},\mathbf{r}_\varphi\rangle \) — a stray conjugate that silently corrupts every downstream adjoint.
• Wrong slot for the representative. Claiming \( \varphi(\mathbf{x}) = \langle \mathbf{r}_\varphi, \mathbf{x}\rangle \) with first-slot-linear convention; this equals \( \overline{\varphi(\mathbf{x})} \), correct only when \( \varphi \) is real-valued.
• Assuming orthonormality is a hypothesis of the theorem. It is a convenience for the formula, not the theorem. With a general basis one must invert the Gram matrix: \( (\mathbf{r}_\varphi)_j = \sum_k (G^{-1})_{jk}\,\overline{\varphi(\mathbf{e}_k)} \).
• Believing \( V^* = V \) as sets. Functionals and vectors are distinct objects; Riesz gives a canonical isomorphism, not equality. Change the inner product and the same \( \varphi \) represents a different \( \mathbf{r}_\varphi \).
• Forgetting positive-definiteness in uniqueness. Verifying existence, skipping step 8, then wrongly assuming uniqueness holds for any bilinear form.

Discussion

The heart of the theorem is that an inner product is exactly the structure needed to turn covectors into vectors. A linear functional lives in the dual \( V^* \); abstractly \( V^* \) is isomorphic to \( V \) merely because they share dimension \( n \) (whence the appeal to basis dimension-invariance), but that isomorphism is basis-dependent and physically meaningless. What Riesz adds is a canonical isomorphism: fix the inner product, and there is exactly one natural way to lower each functional onto a vector. In differential geometry this is the "musical isomorphism" \( \sharp \) that raises indices, \( r^i = g^{ij}\varphi_j \), with the metric \( g \) playing the role of the inner product.

Over \( \mathbb{C} \) the map is a conjugate-linear bijection, not a linear one. This antilinearity is not a blemish — it is precisely why bras and kets behave as they do in quantum mechanics. A ket \( |\psi\rangle \in V \) has a companion bra \( \langle\psi| \in V^* \), and the correspondence \( |\psi\rangle \leftrightarrow \langle\psi| \) is antilinear: \( c|\psi\rangle \leftrightarrow \bar c\langle\psi| \). Dirac's entire notation is a fluent, coordinate-free deployment of the Riesz map, extended via completeness to the infinite-dimensional Hilbert spaces of wavefunctions.

Riesz representation is also the definitional engine behind the adjoint. Given a linear map \( T:V\to V \), the expression \( \mathbf{x}\mapsto \langle T\mathbf{x},\mathbf{y}\rangle \) is, for each fixed \( \mathbf{y} \), a linear functional of \( \mathbf{x} \); Riesz hands us a unique vector, which we define to be \( T^{*}\mathbf{y} \). Thus \( \langle T\mathbf{x},\mathbf{y}\rangle = \langle \mathbf{x},T^{*}\mathbf{y}\rangle \) is well-defined precisely because of this theorem. Self-adjointness \( (T = T^{*}) \), the spectral theorem, and hence the reality of measured eigenvalues all rest on this foundation.

At the deepest level the theorem is a statement about naturality. The assignment \( V \mapsto V^{*} \) is a contravariant functor, and there is no natural isomorphism \( V \cong V^{*} \) of vector spaces alone — any attempt requires choosing a basis and is not natural. Endowing \( V \) with an inner product breaks this obstruction: the pairing \( \langle\cdot,\cdot\rangle \), an element of \( V^{*}\otimes \overline{V}^{*} \), is the extra datum that manufactures the canonical (conjugate-linear-natural) identification. The finite-dimensional Riesz theorem is thus the precise sense in which "having a metric" and "being able to identify vectors with covectors" are one and the same fact.

Common misconceptions. The theorem does not say the dual equals the original space, nor that every vector space is self-dual; without an inner product there is no canonical identification. It also does not extend verbatim to infinite dimensions — there, boundedness of \( \varphi \) and completeness of the space are essential, and unbounded functionals (ubiquitous in quantum mechanics, e.g. position "eigenfunctionals") have no honest Riesz representative inside the space.

Worked examples

Problems

1. In \( \mathbb{R}^3 \) with the dot product, find the Riesz representative of \( \varphi(\mathbf{x}) = x_1 + x_2 + x_3 \) and its norm.
   Solution

   By the formula \( (\mathbf{r}_\varphi)_i = \varphi(\mathbf{e}_i) = 1 \) for each \( i \), so \( \mathbf{r}_\varphi = (1,1,1) \). Norm \( \|\mathbf{r}_\varphi\| = \sqrt{1+1+1} = \sqrt{3} \approx 1.73 \). Check: \( \mathbf{x}\cdot(1,1,1) = x_1+x_2+x_3 = \varphi(\mathbf{x}) \).
2. In \( \mathbb{C}^2 \) with \( \langle\mathbf{a},\mathbf{b}\rangle = a_1\bar b_1 + a_2\bar b_2 \), find \( \mathbf{r}_\varphi \) for \( \varphi(\mathbf{x}) = 2x_1 - i\,x_2 \).
   Solution

   \( \varphi(\mathbf{e}_1)=2,\ \varphi(\mathbf{e}_2)=-i \). Conjugate: \( \overline{2}=2,\ \overline{-i}=i \). Thus \( \mathbf{r}_\varphi = (2,\,i) \). Verify: \( \langle\mathbf{x},(2,i)\rangle = x_1\bar 2 + x_2\bar i = 2x_1 - i x_2 \). Norm \( \sqrt{4+1}=\sqrt5 \).
3. Let \( V=\mathbb{R}^2 \) with the non-standard inner product \( \langle\mathbf{a},\mathbf{b}\rangle = 2a_1b_1 + a_2b_2 \) (positive-definite Gram matrix \( G=\operatorname{diag}(2,1) \)). Find \( \mathbf{r}_\varphi \) for \( \varphi(\mathbf{x}) = x_1 + x_2 \).
   Solution

   The basis \( \{\mathbf{e}_1,\mathbf{e}_2\} \) is not orthonormal here, so use \( \mathbf{r}_\varphi = G^{-1}\boldsymbol{\varphi} \) with \( \boldsymbol{\varphi}=(\varphi(\mathbf{e}_1),\varphi(\mathbf{e}_2))=(1,1) \) and \( G^{-1}=\operatorname{diag}(1/2,1) \). Hence \( \mathbf{r}_\varphi = (1/2,\,1) \). Check: \( \langle\mathbf{x},(1/2,1)\rangle = 2x_1(1/2) + x_2(1) = x_1 + x_2 \). Note it is not \( (1,1) \) — the metric matters.
4. Use Riesz representation to show the adjoint \( T^{*} \) of \( T=\begin{pmatrix}0&1\\2&0\end{pmatrix} \) on real \( \mathbb{R}^2 \) (dot product) is the transpose; compute \( T^{*} \).
   Solution

   For fixed \( \mathbf{y} \), \( \mathbf{x}\mapsto\langle T\mathbf{x},\mathbf{y}\rangle \) is a functional whose Riesz vector is \( T^{*}\mathbf{y} \). Compute \( \langle T\mathbf{x},\mathbf{y}\rangle = (x_2)y_1 + (2x_1)y_2 = x_1(2y_2) + x_2(y_1) = \langle\mathbf{x},(2y_2,\,y_1)\rangle \). So \( T^{*}\mathbf{y}=(2y_2,y_1) \), i.e. \( T^{*}=\begin{pmatrix}0&2\\1&0\end{pmatrix} = T^{\top} \). Over \( \mathbb{R} \) the adjoint is the transpose; over \( \mathbb{C} \) it would be the conjugate transpose.
5. Prove that over \( \mathbb{C} \) the Riesz map \( \varphi\mapsto\mathbf{r}_\varphi \) is conjugate-linear: \( \mathbf{r}_{c\varphi} = \bar c\,\mathbf{r}_\varphi \).
   Solution

   Using the construction, \( \mathbf{r}_{c\varphi} = \sum_i \overline{(c\varphi)(\mathbf{e}_i)}\,\mathbf{e}_i = \sum_i \overline{c\,\varphi(\mathbf{e}_i)}\,\mathbf{e}_i = \bar c\sum_i \overline{\varphi(\mathbf{e}_i)}\,\mathbf{e}_i = \bar c\,\mathbf{r}_\varphi \). The bar on \( c \) is exactly the antilinearity making the ket-to-bra correspondence \( c|\psi\rangle\mapsto\bar c\langle\psi| \) work. Since also \( \mathbf{r}_{\varphi+\psi}=\mathbf{r}_\varphi+\mathbf{r}_\psi \), the map is conjugate-linear.