Let d be a metric in X. Prove that p(x,y)=(d(x,y))/(1+d(x,y)) is also a metric in X.© BrainMass Inc. brainmass.com October 16, 2018, 6:31 pm ad1c9bdddf
From the condition, d is a metric in X. To show that p is a metric in X, we have to verify the followings.
Since d(x,y)>=0, then p(x,y)=d(x,y)/(1+d(x,y))>=0.
(2) p(x,y)=0 if and only if x=y.
If p(x,y)=d(x,y)/(1+d(x,y))=0, then d(x,y)=0, then x=y because d is a metric.
A metrics proof is provided. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.
Real Analysis Textbook Metric Spaces
Please solve the problems in B.pdf file by using the textbook.
The topics covered in this problem set are:
1. Metric spaces. Basic concepts ( Sections 5.1-5.2 )
Problems: - Section 5--# 1,2,4,5,6,7
2. Convergence. Open and closed sets (Sections 6.1-6.6)
Problems: - Section 6--# 1,2,3,4,5,9,10
3. Complete metric spaces (Sections 7.1-7.4)
Problems: - Section 7--# 1,3,4,5,7,9
4. Contraction mappings (Sections 8.1-8.3)
Problems: - Section 8 --# 1,2,4,6
5. Topological spaces
a. Basic Concepts (Sections 9.1,9.3,126.96.36.199)
Problems: - Section 9 --# 1,2,5
b. Compactness (Sections 10.1-10.3)
c. Compactness in Metric spaces (Sections 11.1-11.4)
Problems: - Section 11 --# 1,2,3,4
d. Real functions on Metric and Topological Spaces(Sections 188.8.131.52)
Problems: - Section 12 --# 1,2,3,5,6
6. Linear spaces
a. Basic Concepts (Sections 13..1-13.6)
Problems: - Section 13 --# 1,2,3,5,6View Full Posting Details