Exercise 3. (The triangle inequality) Show that dist (p, g) + dist (g, r) > dist(p,r) for all p, q in R". By a metric on a set X we mean a mapping d: X x X →R such that 1. d(p, q) > 0, with equality if and only if p = q. 2. d(p, q) = d(4,p). %3D "(4'd)p < (4 'b)p + (b 'd)p These properties are called, respectively, positive-definiteness, symmetry, and the triangle inequality. The pair (X, d) is called a metric space. Using the above exercise, one immediately checks that (R", dist) is a metric space. Ge- ometry, in its broadest definition, is the study of metric spaces, and Euclidean Geometry, in the modern sense, is the study of the metric space (R", dist). Finally, we define the angle between a pair of nonzero vectors in R" by angle(p, q) := cos-1 p.q) Note that the above is well defined by the Cauchy-Schwartz inequality. Now we have all the necessary tools to prove the most famous result in all of mathematics:
Exercise 3. (The triangle inequality) Show that dist (p, g) + dist (g, r) > dist(p,r) for all p, q in R". By a metric on a set X we mean a mapping d: X x X →R such that 1. d(p, q) > 0, with equality if and only if p = q. 2. d(p, q) = d(4,p). %3D "(4'd)p < (4 'b)p + (b 'd)p These properties are called, respectively, positive-definiteness, symmetry, and the triangle inequality. The pair (X, d) is called a metric space. Using the above exercise, one immediately checks that (R", dist) is a metric space. Ge- ometry, in its broadest definition, is the study of metric spaces, and Euclidean Geometry, in the modern sense, is the study of the metric space (R", dist). Finally, we define the angle between a pair of nonzero vectors in R" by angle(p, q) := cos-1 p.q) Note that the above is well defined by the Cauchy-Schwartz inequality. Now we have all the necessary tools to prove the most famous result in all of mathematics:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Exercise 3. (The triangle inequality) Show that
dist(p, q) + dist(q, r) > dist(p,r)
for all p, q in R".
By a metric on a set X we mean a mapping d: X × X → R such that
1. d(p, q) > 0, with equality if and only if p = q.
2. d(p, q) = d(q,p).
3. d(p, g) + d(g, r) > d(p,r).
These properties are called, respectively, positive-definiteness, symmetry, and
the triangle inequality. The pair (X, d) is called a metric space. Using the
above exercise, one immediately checks that (R", dist) is a metric space. Ge-
ometry, in its broadest definition, is the study of metric spaces, and Euclidean
Geometry, in the modern sense, is the study of the metric space (R", dist).
Finally, we define the angle between a pair of nonzero vectors in R" by
angle(p, q) := cos-1 (p,q)
Note that the above is well defined by the Cauchy-Schwartz inequality. Now
we have all the necessary tools to prove the most famous result in all of
mathematics:
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