ideas from elementary algebra in this case. Consider the following quadratic polynomial in Since it is nonnegative, it has at most one real root for hence its discriminant is less than or equal to zero. That is, Cn - n-dimensional Complex space If with and (where and ) and if the inner product on the vector space is the canonical complex inner product (defined by where the bar notation is used for complex conjugation), then the inequality may be restated more explicitly as follows: That is, L2 For the inner product space of square-integrable complex-valued functions, the following inequality: The Hölder inequality is a generalization of this. Applications Analysis In any inner product space, the triangle inequality is a consequence of the Cauchy–Schwarz inequality, as is now shown: Taking square roots gives the triangle inequality: The Cauchy–Schwarz inequality is used to prove that the inner product is a continuous function with respect to the topology induced by the inner product itself. Geometry The Cauchy–Schwarz inequality allows one to extend the notion of "angle between two vectors" to any real inner-product space by defining: The Cauchy–Schwarz inequality proves that this definition is sensible, by showing that the right-hand side lies in the interval and justifies the notion that (real) Hilbert spaces are simply generalizations of the Euclidean space. It can also be used to define an angle in complex inner-product spaces, by taking the absolute value or the real part of the right-hand side, as is done when extracting a metric from quantum fidelity. Probability theory Let and be random variables, then the covariance inequality is given by: After defining an inner product on the set of random variables using the expectation of their product, the Cauchy–Schwarz inequality becomes To prove the covariance inequality using the Cauchy–Schwarz inequality, let and then where denotes variance and denotes covariance. Proofs There are many different proofs of the Cauchy–Schwarz inequality other than those given below. When consulting other sources, there are often two sources of confusion. First, some authors define to be linear in the second argument rather than the first. Second, some proofs are only valid when the field is and not This section gives proofs of the following theorem: In all of the proofs given below, the proof in the trivial case where at least one of the vectors is zero (or equivalently, in the case where ) is the same. It