The Schwarz inequality

In its simplest form the Schwarz inequality expressed an obvious relation between the products of magnitudes of two real vectors ci and C2 and their scalar product 

It is less obvious to show that this inequality holds also for complex vectors, provided that the scalar product of two complex vectors e and f is defined by e f. The inequality is of the form 

The contour is closed at z - —oo where the integrand is zero. In fact the integrand has to vanish faster than z as z — oo because the length of the added path diverges like z in that limit. 

Note that in the scalar product e f the order is important, that is, e f = (f e). To prove the inequality (1.81) we start from 

Equation (1.80) can be applied also to real ftmctions that can be viewed as vectors with a continuous ordering index. We can make the identification ca- = (cA ca) /2 k = 1,2 and ci. C2 = /to get 

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This is related to the Schwarz inequality in vector algebra,... 

Proof. [Sketch] Except for the Schwarz inequality, unimaginative calculations suffice for the proof. The Schwarz inequality follows from... 

The triangle inequality follows from the Schwarz inequality. ... 

This is the generalization of Eq. (8), first obtained by Fiirth9 in 1933. Our derivation 10 is, of course, equivalent to his, but perhaps a little more direct by explicit reference to the Schwarz inequality, Eq. (11). 

Alternatively (and equivalently), we can say that if R,), Rj) are any two vectors in the space, with scalar product (R R7), then M is a Euclidean space if, and only if, they satisfy the Schwarz inequality... 

The proof that the criteria (9.27a-c) are indeed equivalent to the Schwarz inequality (9.24), and thus to the other criteria (9.23), (9.26) for a Euclidean space, is sketched in Sidebar 9.3. 

Under this condition, the vectors are co-aligned (with critical angle 0X2c = 0), so the Schwarz inequality (11.4) necessarily becomes an equality... 

The proof of Theorem 19 repeats practically all the steps of Theorem 18. For example, according to formulae (4.7), (4.13), and the Schwarz inequality (4.34), we obtain... 

On the other hand, from the condition (4.35) and the Schwarz inequality, we... 

The absolute value of the first variation of the misfit functional can be estimated using equation (5.8) and the Schwarz inequality (A.38) ... 

An interesting implication of the Schwarz inequality appears in the relationship between averages and correlations involving two observables A and B. Let P be the probability that the system is in state n and let A and B be the values of these observables in this state. Then A = Pn n,... 

The Cauchy inequality [2], also known as the Schwarz inequality, states that for any two vectors a and b... 

In practice, integral screening is employed before the integrals are computed. The screening is performed using the Schwarz inequality... 

Proof. [88] According to the Schwarz inequality, it follows from Eq. (11) that... 

Schwarz inequality. For any two vectors belonging to the Euclidean space the Schwarz inequality holds ... 

The Schwarz inequality agrees with what everyone recalls about the dot product of two vectors ( v) = x lly II cos 6, where 6 is the angle between the two vectors. Taking the absolute value of both sides, we obtain (. v) = x v eos6 r v. ... 

After the new inner product definition is introduced, related quantities the length of a vector and the distance between the vectors are defined in exactly the same way as in the Euclidean space. Also the definitions of orthogonality and of the Schwarz inequality remain unchanged. 

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