In algebra, the Cauchy-Schwarz Inequality, also known as the Cauchy–Bunyakovsky–Schwarz Inequality or informally as Cauchy-Schwarz, is an inequality with many ubiquitous formulations in abstract algebra, calculus, and contest mathematics. In high-school competitions, its applications are limited to elementary and linear algebra.
Its elementary algebraic formulation is often referred to as Cauchy's Inequality and states that for any list of reals and , with equality if and only if there exists a constant such that for all , or if one list consists of only zeroes. Along with the AM-GM Inequality, Cauchy-Schwarz forms the foundation for inequality problems in intermediate and olympiad competitions. It is particularly crucial in proof-based contests.
Its vector formulation states that for any vectors and in , where is the dot product of and and is the norm of , with equality if and only if there exists a scalar such that , or if one of the vectors is zero. This formulation comes in handy in linear algebra problems at intermediate and olympiad problems.
The full Cauchy-Schwarz Inequality is written in terms of abstract vector spaces. Under this formulation, the elementary algebraic, linear algebraic, and calculus formulations are different cases of the general inequality.
Here is a list of proofs of Cauchy-Schwarz.
Consider the vectors and . If is the angle formed by and , then the left-hand side of the inequality is equal to the square of the dot product of and , or .The right hand side of the inequality is equal to . The inequality then follows from , with equality when one of is a multiple of the other, as desired.
The inequality sometimes appears in the following form.
Let and be complex numbers. Then This appears to be more powerful, but it follows from
A Useful Inequality
Also known as Sedrakyan's Inequality, Bergström's Inequality, Engel's Form or Titu's Lemma the following inequality is a direct result of Cauchy-Schwarz inequality:
For any real numbers and where the following is true:
Real Vector Spaces
Let be a vector space, and let be an inner product. Then for any , with equality if and only if there exist constants not both zero such that . The following proofs assume the inner product to be real-valued and commutative, and so only apply to vector spaces over the real numbers.
Consider the polynomial of This must always be greater than or equal to zero, so it must have a non-positive discriminant, i.e., must be less than or equal to , with equality when or when there exists some scalar such that , as desired.
We consider Since this is always greater than or equal to zero, we have Now, if either or is equal to , then . Otherwise, we may normalize so that , and we have with equality when and may be scaled to each other, as desired.
Consider for some scalar . Then: (by the Trivial Inequality) . Now, let . Then, we have: .
Complex Vector Spaces
For any two vectors in the complex vector space , the following holds: with equality holding only when are linearly dependent.
The following proof, a geometric argument that uses only the algebraic properties of the inner product, was discovered by Tarung Bhimnathwala in 2021.
Define the unit vectors , as and . Put . In other words, is the complex argument of and lies on the unit circle. Let and . Importantly, we have Since and , this calculation shows that and form an orthogonal basis of the linear subspace spanned by and . Thus we can think of and as lying on the unit sphere in this subspace, which is isomorphic to . Another thing to note is that
The previous two calculations established that and are orthogonal, and that the sum of their squared norms is . Now we have Equality holds when either or , or equivalently when and . Lastly, multiplying each side by , we have
- Consider the function , where is a positive integer. Show that . (Source)
- (APMO 1991 #3) Let , , , , , , , be positive real numbers such that . Show that
- Let be a triangle such that
where and denote its semiperimeter and inradius, respectively. Prove that triangle is similar to a triangle whose side lengths are all positive integers with no common divisor and determine those integers. (Source)
- is a point inside a given triangle . are the feet of the perpendiculars from to the lines , respectively. Find all for which
- The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele.
- Problem Solving Strategies by Arthur Engel contains significant material on inequalities.