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Corollary 2.53 Let (M, d) be a metric space and K C M. The following statements are equivalent: (i) K is relatively compact. (ii) K is totally bounded and (K,d) is complete.

Corollary 2.53 Let (M, d) be a metric space and K C M. The following statements are equivalent: (i) K is relatively compact. (ii) K is totally bounded and (K,d) is complete.

Advanced Engineering 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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prove corollary

Corollary 2.53 Let (M, d) be a metric space and K C M. The following statements are equivalent: (i) K is relatively compact. (ii) K is totally bounded and (K,d) is complete. Corollary 2.54 Let (M, d) be a complete metric space and K C M. The following statements are equivalent: (i) K is relatively compact. (ii) K is totally bounded. Proof: excercises.
Transcribed Image Text:Corollary 2.53 Let (M, d) be a metric space and K C M. The following statements are equivalent: (i) K is relatively compact. (ii) K is totally bounded and (K,d) is complete. Corollary 2.54 Let (M, d) be a complete metric space and K C M. The following statements are equivalent: (i) K is relatively compact. (ii) K is totally bounded. Proof: excercises.
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