Problems 1.1 | d(x,y) = |x| + |y| Metric Space Chapter 01 | Functional Analysis Kreyszig
Problems 1.1 | d(x,y) = |x| + |y| Metric Space Chapter 01 | Functional Analysis Kreyszig
d(x,y) = |x| + |y| | Question No 6 | Problems 1.1 | Metric Space | Chapter 01 | Problems 1.1 | Introductory Functional Analysis with Applications | Erwin Kreyszig
d(x,y) = |x| + |y|
Book Name : Introductory Functional Analysis with Applications
By : Erwin Kreyszig
Chapter Number : 01
Chapter Name : Metric Space
Lecture Number : 9
By (Name) : Awais Rasool
Exercise Number : 1.1
Problems Number: 1.1
Question Number : 09
Part Number : 0
Example Number : 03
Awais Rasool Shah
Topics Name :
Chapter 1. Metric Spaces
1.1 Metric Space
1.2 Further Examples of Metric Spaces
1.3 Open Set, Closed Set, Neighborhood
1.4 Convergence, Cauchy Sequence, Completeness
1.5 Examples. Completeness Proofs
1.6 Completion of Metric Spaces
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Definition (Metric space , Metric).
Consider a non-empty set 𝑥 and a function 𝑑 : 𝑥∗𝑥 𝑅^+ "∪{0}".
This function 𝑑 is called metric on 𝑥 if following conditions are holds:
1) 𝑑(𝑥,𝑦)≥0
2) 𝑑(𝑥,𝑦)=0 𝑖𝑓𝑓 𝑥=𝑦
3) 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) (Symmetry)
4) 𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦)
The set X with d is called metric Space and written as (X, 𝑑).
Problems 1.1 | d(x,y) = |x| + |y| Metric Space Chapter 01 | Functional Analysis Kreyszig
Show that 𝑑(𝑥,𝑦)=|𝑥|+|𝑦| is a metric on 𝑅, ∀ 𝑥, 𝑦∈𝑅.
Sol:
𝑑(𝑥,𝑦)≥0
𝑑(𝑥,𝑦)=|𝑥|+|𝑦|≥0
The absolute value of any real number is always non-negative and
the sum of a non-negative number is also non-negative , so this
property is satisfied.
𝑑(𝑥,𝑦)=0 𝑖𝑓𝑓 𝑥=𝑦
𝑑(𝑥,𝑦)=|𝑥|+|𝑦|=0
Since both terms are non-negative, each term must individually
be zero:
"⟺ " |𝑥|=0 "⟺ " |𝑦|=0
"⟺" 𝑥=0 "⟺" 𝑦=0
"⟺" 𝑥=𝑦
𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) (Symmetry)
𝑑(𝑥,𝑦)=|𝑥|+|𝑦|
=|𝑦|+|𝑥|
=𝑑(𝑦,𝑥)
𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦)
𝑑(𝑥,𝑦)=|𝑥|+|𝑦|
=|𝑥+𝑧−𝑧|+|𝑦+𝑧−𝑧|
≤|𝑥+𝑧|+|−𝑧|+|𝑦+𝑧|+|−𝑧|
Using triangle inequality |𝑎+𝑏|≤|𝑎|+|𝑏|
≤|𝑥+𝑧|+|𝑧|+|𝑦+𝑧|+|𝑧|
=|𝑥+𝑧|+2|𝑧|+|𝑦+𝑧|
≠ 𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦)
Since the function 𝑑(𝑥,𝑦)=|𝑥|+|𝑦| not satisfies the four properties of a metric space.