CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Finite-dimensional representations separate points

Statement

Assume the Axiom of Choice. For distinct elements $x \neq y$ of a compact Lie group $G$ there is a finite-dimensional unitary representation $π$ with $π (x) \neq π (y)$ .

Facts & Assumptions

Given: Assume the Axiom of Choice, a compact Lie group $G$ , and distinct points $x, y \in G$ .

[A1]

The Axiom of Choice is The Axiom of Choice; it enters through the Haar theory behind [L1].

[L1]

Finite linear combinations of matrix coefficients of finite-dimensional unitary representations are uniformly dense in $C (G)$ (Matrix coefficients are uniformly dense in C(G)).

[L2]

A Lie group is Hausdorff, so $G$ is compact Hausdorff. Under dependent choice, disjoint closed subsets of a compact Hausdorff space are separated by a continuous function into $[0, 1]$ ; the assumed Axiom of Choice supplies dependent choice (Lie group, Under dependent choice a compact Hausdorff space is Tychonoff, and its disjoint closed sets are separated by continuous functions, AC supplies countable selections and prescribed serial paths).

Proof

technique · direct

1.1

The singletons ${x}$ and ${y}$ are disjoint closed subsets of the compact Hausdorff space $G$ . By [L2] choose $f \in C (G, [0, 1])$ with $f (x) = 0$ and $f (y) = 1$ , and set $η : = ∣ f (x) - f (y) ∣ = 1$ .

2.1

By [L1] choose a finite linear combination $s$ of matrix coefficients with $∥ s - f ∥_{\infty} < η / 4$ ; then $∣ s (x) - s (y) ∣ \geq ∣ f (x) - f (y) ∣ - 2 ∥ s - f ∥_{\infty} > η / 2 > 0$ , so some matrix coefficient $π_{i j}$ occurring in $s$ satisfies $π_{i j} (x) \neq π_{i j} (y)$ .

L1step 1.1

3.1

For that matrix coefficient, with $π$ unitary in an orthonormal basis, $π_{i j} (x) = ⟨ π (x) e_{j}, e_{i} ⟩ \neq ⟨ π (y) e_{j}, e_{i} ⟩ = π_{i j} (y)$ , so $π (x) \neq π (y)$ ; hence the finite-dimensional unitary representation $π$ separates $x$ from $y$ .

A1step 2.1∎

Depends on

Used by

Every compact Lie group is a closed matrix group Corollary

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)