False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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FALSE: Every orthogonal set is an orthonormal basis

Statement

False claim. Every orthogonal set in an inner product space is an orthonormal basis.

Facts & Assumptions

Given: The singleton list containing $(2, 0)$ in standard $R^{2}$ .

[L1]

An orthogonal list only requires distinct members to be pairwise orthogonal; an orthonormal basis additionally requires unit norms and spanning (Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases).

[L2]

In standard $R^{2}$ , the squared norm is the sum of coordinate squares (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

Refutation

technique · counterexample

1.1

A singleton is orthogonal because it has no distinct pair to check. But [L2] gives $∥ (2, 0) ∥ = 2$ , and its span is only a line in $R^{2}$ .

2.1

Thus it is neither normalised nor a basis, exposing both missing requirements in [L1]. The empty orthogonal list supplies the same spanning warning in every nonzero space.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 38 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Sheldon Axler, Linear Algebra Done Right, 4th ed., definitions 6.22 and 6.27 (standard reference, not scraped)