1.Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality states that for all vectors 
and 
of an inner product space it is true that
is the inner product.
Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as：
Moreover, the two sides are equal if and only if 
and 
are linearly dependent (meaning they are parallel: one of the vector's magnitudes is zero, or one is a scalar multiple of the other).
If 
and 
, and the inner product is the standard complex inner product, then the inequality may be restated more explicitly as follows (where the bar notation is used for complex conjugation):
or 