---
tag: 7MN6S22S2Q
kind: proof
name: Proof of the Riesz representation theorem in finite dimensions
msc: [46C05, 15A04]
proves: 04GYPXDXNH
uses: [0VGSYBWHV2, 01EK923SKP, 0RT96J4D3H, 0Y85GYEGSF, 0FWJ0VW561, 09DZ8CD1AK]
sources:
  - url: https://linear.axler.net/LADR4e.pdf
    locator: Theorem 6.42
---

## Proof

We build $y$ from an orthonormal basis, and then prove uniqueness with $x = y - y'$.

1. Let $V$ be a finite-dimensional [[inner product space|01EK923SKP]] over $\mathbb{K} = \R$ or $\C$, with [[inner product|0VGSYBWHV2]] $\ip{\cdot}{\cdot}$ linear in the first argument. Let $f \colon V \to \mathbb{K}$ be a [[linear form|0RT96J4D3H]].
2. For all $u, w \in V$ and $a \in \mathbb{K}$, conjugate symmetry and linearity in the first argument give $\ip{u}{a w} = \overline{a} \ip{u}{w}$. In the same way, $\ip{u}{\cdot}$ is additive.
3. Let $e_1, \dots, e_n$ be an [[orthonormal basis|0Y85GYEGSF]] of $V$. Set $y = \sum_{i=1}^{n} \overline{f(e_i)}\, e_i$.
4. Let $x \in V$. By the [[formula for coordinates|09DZ8CD1AK]], $x = \sum_{i=1}^{n} \ip{x}{e_i} e_i$.
5. Since $f$ is linear, step 4 gives $f(x) = \sum_{i=1}^{n} \ip{x}{e_i} f(e_i)$.
6. By step 2, $\ip{x}{y} = \sum_{i=1}^{n} \overline{\overline{f(e_i)}} \ip{x}{e_i} = \sum_{i=1}^{n} f(e_i) \ip{x}{e_i}$.
7. Steps 5 and 6 give $f(x) = \ip{x}{y}$ for all $x \in V$. This proves existence.
8. For uniqueness, let $y, y' \in V$ with $\ip{x}{y} = f(x) = \ip{x}{y'}$ for all $x \in V$.
9. By step 2, $\ip{x}{y - y'} = \ip{x}{y} - \ip{x}{y'} = 0$ for all $x \in V$.
10. Take $x = y - y'$. Then $\ip{y - y'}{y - y'} = 0$, and positivity of the inner product gives $y - y' = 0$. So $y = y'$.

## Idea

A linear form is fixed by its values on a basis. For an orthonormal basis, the form $x \mapsto \ip{x}{y}$ takes the value $\overline{c_i}$ at $e_i$ when $y = \sum_i c_i e_i$. So we choose $c_i = \overline{f(e_i)}$. Uniqueness holds because only $0$ is orthogonal to every vector.