Schwarz inequality

Thus the average cost per share for John is the arithmetic mean of pi, po,. . . , pn, whereas that for Mary is the harmonic mean of these n numbers. Since the harmonic mean is less than or equal to the arithmetic mean for any set of positive numbers and the two means are equal only i pi=po = =pn, we conclude that the average cost per share for Mary is less than that for John if two of the prices Pi are distinct. One can also give a proof based on the Gaiichy-Schwarz inequality, To this end, define the vectors... 

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

Somewhat weaker conditions are obtained by summing the Cauchy-Schwarz inequalities over r, 5, for example. 

Then INI is a norm. It is often called the norm associated to the scalar product ( , ). Furthermore, we have the so-called Schwarz inequality. ifx, y e V then N,y> < x y. ... 

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. ... 

Also known as the Cauchy-Bunyakovskii-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. 

Problem Prove that the mathematical criteria (9.27a-c) for a proposed scalar product (R R7) are sufficient for the general Schwarz inequality (9.24) in the space M, thereby guaranteeing that M is Euclidean. 

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) ... 

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