CauchySchwarztétel

The CauchySchwarz inequality, also known as the Cauchy-Bunyakovsky-Schwarz inequality, is a fundamental inequality in mathematics. It relates the inner product of two vectors in an inner product space to the product of their norms. In its simplest form, for any two real numbers x and y, the inequality states that (x*x + y*y)(a*a + b*b) >= (xa + yb)*(xa + yb).

More generally, for vectors u and v in an inner product space, the Cauchy-Schwarz inequality is expressed as |⟨u, v⟩|^2 <= ⟨u, u⟩⟨v, v⟩, where ⟨u, v⟩ denotes the inner product of u and v. This can also be written as |⟨u, v⟩| <= ||u|| ||v||, where ||u|| and ||v|| are the norms of vectors u and v respectively.

The equality holds if and only if the vectors u and v are linearly dependent, meaning one is a scalar multiple of the other. This theorem has widespread applications in various fields of mathematics, including linear algebra, functional analysis, probability theory, and physics. It is often used to prove other inequalities and to establish convergence properties. The inequality itself is named after Augustin-Louis Cauchy, Hermann Schwarz, and Viktor Bunyakovsky.