The Cauchy–Schwarz inequality caps how much two vectors can agree
Multiply two vectors together with a dot product and the result can never exceed the product of their lengths. That modest-sounding cap, the Cauchy–Schwarz inequality, took three mathematicians and nearly seven decades to state in full, and it ranks among the most heavily used inequalities anywhere in mathematics.
In plain terms, the inequality bounds the size of an inner product, a generalised dot product, by the lengths of the two vectors involved. Equality happens only when one vector is a scalar multiple of the other. Because inner products can represent finite sums, infinite series or integrals, a single statement covers all three settings at once.
Its history follows those settings. Augustin-Louis Cauchy published the version for sums in 1821. The counterpart for integrals appeared from Viktor Bunyakovsky in 1859 and again from Hermann Schwarz in 1888, and it is Schwarz's argument that forms the modern proof. That shared credit explains the longer label Cauchy–Bunyakovsky–Schwarz.
The flat plane gives the clearest picture. There the dot product of two arrows equals their lengths multiplied together and then by the cosine of the angle between them. A cosine squared can reach at most 1, and it does so exactly when the arrows point the same way or directly opposite. So the inequality is really saying that two directions agree most when they are aligned. Written in coordinates, the squared sum of paired products never exceeds the product of the two sums of squares, and the same statement holds in any number of dimensions.
Ordinary school algebra is enough to prove that coordinate form. One route shows that the gap between the two sides is a sum of squares and so cannot be negative. Another builds a quadratic that never dips below zero; it therefore has at most one real root, forcing its discriminant to be zero or less, which is the inequality itself. A useful offshoot, known as Sedrakyan's lemma, Engel's form or Titu's lemma, handles sums of fractions whose numerators are perfect squares.
Source: Cauchy–Schwarz inequality
