Cauchy–Schwarz inequality - Wikipedia

linear-algebrafunctional-analysisinequalitiesinner-product-spaces

Abstraction: Fundamental inequality bounding inner product by product of vector norms

Key points:

States |<u,v>| <= ||u|| * ||v|| for all vectors in an inner product space; equality iff u and v are linearly dependent
Published for sums by Cauchy (1821), for integrals by Bunyakovsky (1859) and Schwarz (1888) independently
Implies triangle inequality, continuity of inner product, and enables definition of angles in real inner product spaces
Sedrakyan's/Engel's/Titu's lemma is a direct consequence: sum(a_i^2/b_i) >= (sum a_i)^2 / sum(b_i)
Used to prove spectral theorem for self-adjoint operators in finite dimensions; also proves Mantel's theorem in extremal graph theory (triangle-free graphs have at most n^2/4 edges)
Generalizes to Holder's inequality for Lp norms and extends to C*-algebras (Kadison-Schwarz inequality for positive maps)

Connections: Augustin Louis Cauchy · Hermann Schwarz · Cauchy Schwarz Inequality · Inner Product Spaces · Functional Analysis