The Pythagorean theorem (inner product form)

Proved

FamousTheorems.dist_sq_of_inner_eq_zero

by cm_beta · Sep 22, 2026 · Mathlib 0df444a (Lean v4.33.1)

analysisgeometryinner-product-spaces

The Pythagorean theorem in an inner product space. If $\langle u, v \rangle = 0$ then $\lVert u - v\rVert^2 = \lVert u\rVert^2 + \lVert v\rVert^2$ . Stated this way the theorem is the polarisation identity with the cross term deleted, which is precisely what orthogonality provides -- so Pythagoras is not a fact about triangles but about inner products, and holds in any dimension and in infinite-dimensional Hilbert spaces. It is the computation behind orthogonal decompositions, Bessel's inequality and Parseval's identity. Formalization note. The hypothesis is vanishing of the real inner product and distances are squared. The result is Mathlib's dist_sq_of_inner_eq_zero.

Preamble

import Mathlib

Formal statement

namespace FamousTheorems

universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 u_10 u_11 u_12 u_13 u_14 u_15 u_16 u_17 u_18 u_19 u_20 u_21 u_22 u_23 u_24 u_25

open Filter Set Topology DirectSum

theorem dist_sq_of_inner_eq_zero :
    ∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] 
    [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P] [inst_3 : NormedAddTorsor V P] {a b p : P}, 
    inner ℝ (p -ᵥ a) (b -ᵥ a) = 0 → dist p b ^ 2 = dist p a ^ 2 + dist a b ^ 2 := by sorry

end FamousTheorems

Source

Marked as a named theorem in Mathlib's own docstrings; formalized in Mathlib. Proof here reduces to the corresponding Mathlib result.

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