Any of several related inequalities developed by Augustin-Louis Cauchy (Cauchy, Augustin-Louis, Baron) and, later, Herman Schwarz (1843–1921). The inequalities arise from assigning a real number measurement, or norm, to the functions, vectors, or integrals within a particular space in order to analyze their relationship. For functions f and g, whose squares are integrable and thus usable as a norm, (∫fg)²≤(∫f²)(∫g²). For vectors a = (a₁, a₂, a₃,…, a_n) and b = (b₁, b₂, b₃,…, b_n), together with the inner product (see inner product space) for a norm, (Σ(a_i, b_i))² ≤ Σ(a_i)²Σ(b_i)². In addition to functional analysis, these inequalities have important applications in statistics and probability theory.