Cauchy inequality was obtained when Cauchy, a great mathematician, studied the problem of "flow number" in mathematical analysis. But historically, this inequality should be called Cauchy-Bunyakovski-Schwartz inequality, and the high school formula of Cauchy inequality is as follows.

1, general form

(∑ai^2)(∑bi^2)≥(∑ai bi)^2。

The equal sign holds as follows: a1:b1= a2: B2 = … = an: bn, or both ai and bi are zero.

2. Two-dimensional form

(a^2+b^2)(c^2 + d^2)≥(ac+bd)^2。

Equal sign condition: ad=bc.

3. Vector form

|α||β|≥|α β|，α=(a 1，a2，…，an)，β=(b 1，b2，…，bn)(n∈N，n≥2)。

The condition of equal sign is that β is zero vector, or α = λ β (λ∈ r).

4. Triangle type

√(a^2+b^2)+√(c^2+d^2)≥√[(a-c)^2+(b-d)^2]。

Equal sign condition: ad=bc.