Page 88 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Self Assessment

Fill in the blanks:

1. A completion of metric space M is ........................... N s.t. M dense subset of N.
2. Any metric M can be .............................. into complete metric space.

3. Metric M totally ....................... if "  > 0  finite set F  M s.t. M   x F B(x, ) .

4. Subspace M of ................ is totally bounded iff "  > 0  finite H  N s.t. M   z H B(z, ) .

6.6 Summary

 Complete subspace S of metric M is closed.
 Closed subset S of complete M is complete.

 " S B(S) of bdd functions S   with sup norm is complete.
0
 A sequence (x )  M is Cauchy iff  sequence   0 s.t.  ¾¾¾ and d(x , x )  for m
n n n n m n n
> n.

 (x )  M sequence s.t.   0 with å n 1  <  and d(x , x )  " n. Then (x ) is Cauchy.
=
n n n n n+1 n n
6.7 Keywords

Cauchy Sequence: Metric M is complete if every Cauchy sequence in M converges (to a point of M).

Cauchy: A sequence (x )  M is Cauchy iff  sequence   0 s.t.  ¾¾¾ and d(x , x )  for
0
n n n n m n n
m > n.
Completion: Any metric space M has a completion.

6.8 Review Questions

1. Define Completeness.

2. Discuss the Cauchy.
3. Explain contraction mapping theorem.

4. Describe total boundness.
Answers: Self Assessment

1. Complete metric space 2. isometrically embedded
3. bounded 4. metric N

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