Cauchy-Schwarz inequality

This result is an application of the Cauchy-Schwarz inequality.

But, we see that and by the Cauchy-Schwarz inequality, .

To do so, we first recognize a form of the Cauchy-Schwarz inequality:

We integrate over and apply the Cauchy-Schwarz inequality, to obtain:

The Cauchy-Schwarz inequality is met with equality when the two vectors involved are collinear.

The fact that the above sum converges for every x follows from the Cauchy-Schwarz inequality.

Which is true by the Cauchy-Schwarz inequality.

For any legal operators the following inequality, a consequence of the Cauchy-Schwarz inequality, is true.

It is a corollary of the Cauchy-Schwarz inequality that the correlation cannot exceed 1 in absolute value.

Using the Cauchy-Schwarz inequality and the inequality (1), we get:

This can be shown by an application of the Cauchy-Schwarz inequality to the definition of .

In cases where the ideal linear system assumptions are insufficient, the Cauchy-Schwarz inequality guarantees a value of .

The Cauchy-Schwarz inequality is usually used to show Bessel's inequality.

It follows, essentially from the Cauchy-Schwarz inequality, that the Fourier series of f is absolutely summable.

The Cauchy-Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion.

Here the sum also has only countably many nonzero terms, and is unconditionally convergent by the Cauchy-Schwarz inequality.

A firmly non-expansive mapping is always non-expansive, via the Cauchy-Schwarz inequality.

This inequality depends on the dimension n of the underlying vector space and follows directly from the Cauchy-Schwarz inequality.

When Φ is 2-positive, a stronger assumption than merely positive, one has something that looks very similar to the original Cauchy-Schwarz inequality: "

It uses only the arc length formula, expression for the area of a plane region from Green's theorem, and the Cauchy-Schwarz inequality.

An analogue to the Cauchy-Schwarz inequality holds for an inner-product A-module E:

One can see from the proof that the Cauchy-Schwarz inequality is a consequence of the "positivity" and "anti-symmetry" inner-product axioms.

For f and g in L, this integral exists because of the Cauchy-Schwarz inequality, and defines an inner product on the space.

The Paley-Zygmund inequality is sometimes used instead of the Cauchy-Schwarz inequality and may occasionally give more refined results.

The geodesic minimizes the entropy, due to the Cauchy-Schwarz inequality, which states that the action is bounded below by the length of the curve, squared.