Holders_Inequality_for_Sums

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Holder's inequality for sums: for conjugate exponents p, q > 1, the inner product is bounded by the product of Lp norms.

Why this node was retired

The posted statement is

theorem Holders_Inequality_for_Sums (n : ℕ) (p q : ℝ) (hp : 1 < p) (hpq : 1 / p + 1 / q = 1) (a b : Fin n → ℝ) (ha : ∀ i, 0 ≤ a i) (hb : ∀ i, 0 ≤ b i) : ∑ i : Fin n, a i * b i ≤ (∑ i : Fin n, a i ^ (2 : ℕ)) * (∑ i : Fin n, b i ^ (2 : ℕ)) := by sorry

The alleged Hölder/Cauchy–Schwarz bound omits the 1/p and 1/q powers on the two norm sums.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For p,q>1 with 1/p+1/q=1, prove ∑|aᵢbᵢ|≤(∑|aᵢ|ᵖ)^(1/p)(∑|bᵢ|ᑫ)^(1/q), with endpoint cases treated separately. For p=q=2 restore both square roots.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.