Page 275 - Matrix Analysis & Applied Linear Algebra

5.1 Vector Norms 271
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• The distance between vectors in can be visualized with the aid of the
n n
parallelogram law as shown in Figure 5.1.2, so for vectors in and C ,
the distance between u and v is naturally defined to be u − v .
u

u - v ||u - v|| v

Figure 5.1.2

Standard Inner Product
The scalar terms defined by

n n

T
∗
x y = x i y i ∈ and x y = ¯ x i y i ∈C
i=1 i=1
n n
are called the standard inner products for and C , respectively.
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The Cauchy–Bunyakovskii–Schwarz (CBS) inequality is one of the most
important inequalities in mathematics. It relates inner product to norm.

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The Cauchy–Bunyakovskii–Schwarz inequality is named in honor of the three men who played
a role in its development. The basic inequality for real numbers is attributed to Cauchy in 1821,
whereas Schwarz and Bunyakovskii contributed by later formulating useful generalizations of
the inequality involving integrals of functions.
Augustin-Louis Cauchy (1789–1857) was a French mathematician who is generally regarded
as being the founder of mathematical analysis—including the theory of complex functions.
Although deeply embroiled in political turmoil for much of his life (he was a partisan of the
Bourbons), Cauchy emerged as one of the most prolific mathematicians of all time. He authored
at least 789 mathematical papers, and his collected works fill 27 volumes—this is on a par with
Cayley and second only to Euler. It is said that more theorems, concepts, and methods bear
Cauchy’s name than any other mathematician.
Victor Bunyakovskii (1804–1889) was a Russian professor of mathematics at St. Petersburg, and
in 1859 he extended Cauchy’s inequality for discrete sums to integrals of continuous functions.
His contribution was overlooked by western mathematicians for many years, and his name is
often omitted in classical texts that simply refer to the Cauchy–Schwarz inequality.
Hermann Amandus Schwarz (1843–1921) was a student and successor of the famous German
mathematician Karl Weierstrass at the University of Berlin. Schwarz independently generalized
Cauchy’s inequality just as Bunyakovskii had done earlier.