Difference between revisions of "Complete Metric Spaces"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
(Created page with "<h1 id="toc0"><span>Complete Metric Spaces</span></h1> <p>Recall that a sequence <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> in <span class="math-inline...")
 
 
Line 1: Line 1:
−
<h1 id="toc0"><span>Complete Metric Spaces</span></h1>
 
 
<p>Recall that a sequence <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> in <span class="math-inline"><math>M</math></span> is called a Cauchy sequence if for all <span class="math-inline"><math>\epsilon > 0</math></span> there exists an <span class="math-inline"><math>N \in \mathbb{N}</math></span> such that if <span class="math-inline"><math>m, n \geq N</math></span> then <span class="math-inline"><math>d(x_m, x_n) < \epsilon</math></span>.</p>
 
<p>Recall that a sequence <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> in <span class="math-inline"><math>M</math></span> is called a Cauchy sequence if for all <span class="math-inline"><math>\epsilon > 0</math></span> there exists an <span class="math-inline"><math>N \in \mathbb{N}</math></span> such that if <span class="math-inline"><math>m, n \geq N</math></span> then <span class="math-inline"><math>d(x_m, x_n) < \epsilon</math></span>.</p>
 
<p>Consider any metric space <span class="math-inline"><math>(M, d)</math></span>. If <span class="math-inline"><math>(M, d)</math></span> is such that every Cauchy sequence converges in <span class="math-inline"><math>M</math></span>, then we give this metric space a special name.</p>
 
<p>Consider any metric space <span class="math-inline"><math>(M, d)</math></span>. If <span class="math-inline"><math>(M, d)</math></span> is such that every Cauchy sequence converges in <span class="math-inline"><math>M</math></span>, then we give this metric space a special name.</p>
Line 8: Line 7:
 
<div style="text-align: center;"><math>\begin{align} \quad (x_n)_{n=1}^{\infty} = \left ( 1 - \frac{1}{n} \right )_{n=1}^{\infty} = \left ( 0, \frac{1}{2}, \frac{2}{3}, ... \right ) \end{align}</math></div>
 
<div style="text-align: center;"><math>\begin{align} \quad (x_n)_{n=1}^{\infty} = \left ( 1 - \frac{1}{n} \right )_{n=1}^{\infty} = \left ( 0, \frac{1}{2}, \frac{2}{3}, ... \right ) \end{align}</math></div>
 
<p>We claim that <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> is a Cauchy sequence. Let's prove this. Let <span class="math-inline"><math>m, n \in \mathbb{N}</math></span>, and consider:</p>
 
<p>We claim that <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> is a Cauchy sequence. Let's prove this. Let <span class="math-inline"><math>m, n \in \mathbb{N}</math></span>, and consider:</p>
−
<div style="text-align: center;"><math>\begin{align} \quad d(x_m, x_n) = \biggr \lvert \left ( 1 - \frac{1}{m} \right ) - \left ( 1 - \frac{1}{n} \right ) \biggr \rvert = \biggr \lvert \frac{1}{n} - \frac{1}{m} \biggr \rvert \leq \biggr \lvert \frac{1}{n} \biggr \rvert + \biggr \lvert \frac{1}{m} \biggr \rvert = \frac{1}{n} + \frac{1}{m} \end{align}</math></div>
+
<div style="text-align: center;"><math>\begin{align} \quad d(x_m, x_n) = \left| \left ( 1 - \frac{1}{m} \right ) - \left ( 1 - \frac{1}{n} \right ) \right| = \left| \frac{1}{n} - \frac{1}{m} \right| \leq \left| \frac{1}{n} \right| + \left| \frac{1}{m} \right| = \frac{1}{n} + \frac{1}{m} \end{align}</math></div>
 
<p>Choose <span class="math-inline"><math>N</math></span> such that <span class="math-inline"><math>N > \frac{2}{\epsilon}</math></span>. Then if <span class="math-inline"><math>m, n \in \mathbb{N}</math></span> are such that <span class="math-inline"><math>m, n, \geq N</math></span> then:</p>
 
<p>Choose <span class="math-inline"><math>N</math></span> such that <span class="math-inline"><math>N > \frac{2}{\epsilon}</math></span>. Then if <span class="math-inline"><math>m, n \in \mathbb{N}</math></span> are such that <span class="math-inline"><math>m, n, \geq N</math></span> then:</p>
 
<div style="text-align: center;"><math>\begin{align} \quad m \geq N > \frac{2}{\epsilon} \quad \mathrm{so} \quad \frac{1}{m} \leq \frac{1}{N} < \frac{\epsilon}{2} \\ \quad n \geq N > \frac{2}{\epsilon} \quad \mathrm{so} \quad \frac{1}{n} \leq \frac{1}{N} < \frac{\epsilon}{2} \end{align}</math></div>
 
<div style="text-align: center;"><math>\begin{align} \quad m \geq N > \frac{2}{\epsilon} \quad \mathrm{so} \quad \frac{1}{m} \leq \frac{1}{N} < \frac{\epsilon}{2} \\ \quad n \geq N > \frac{2}{\epsilon} \quad \mathrm{so} \quad \frac{1}{n} \leq \frac{1}{N} < \frac{\epsilon}{2} \end{align}</math></div>
Line 14: Line 13:
 
<div style="text-align: center;"><math>\begin{align} \quad d(x_m, x_n) \leq \frac{1}{n} + \frac{1}{m} < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align}</math></div>
 
<div style="text-align: center;"><math>\begin{align} \quad d(x_m, x_n) \leq \frac{1}{n} + \frac{1}{m} < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align}</math></div>
 
<p>So <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> is indeed a Cauchy sequence. However, it should be intuitively clear that <span class="math-inline"><math>\lim_{n \to \infty} x_n = 1</math></span>, but <span class="math-inline"><math>1 \not \in [0, 1)</math></span>! Therefore <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> does not converge in <span class="math-inline"><math>M</math></span> and hence <span class="math-inline"><math>(M, d)</math></span> is not a complete metric space.</p>
 
<p>So <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> is indeed a Cauchy sequence. However, it should be intuitively clear that <span class="math-inline"><math>\lim_{n \to \infty} x_n = 1</math></span>, but <span class="math-inline"><math>1 \not \in [0, 1)</math></span>! Therefore <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> does not converge in <span class="math-inline"><math>M</math></span> and hence <span class="math-inline"><math>(M, d)</math></span> is not a complete metric space.</p>
 +
  
 
==Licensing==
 
==Licensing==
 
Content obtained and/or adapted from:
 
Content obtained and/or adapted from:
 
* [http://mathonline.wikidot.com/complete-metric-spaces Complete Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license
 
* [http://mathonline.wikidot.com/complete-metric-spaces Complete Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license

Latest revision as of 15:27, 8 November 2021

Recall that a sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_n)_{n=1}^{\infty}} in is called a Cauchy sequence if for all there exists an such that if then .

Consider any metric space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} is such that every Cauchy sequence converges in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , then we give this metric space a special name.

Definition: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} be a metric space. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} is said to be Complete if every Cauchy sequence converges in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} .

In general, it is much easier to show that a metric space is not complete by finding a Cauchy sequence that does not converge in the space. For example, consider the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = [0, 1)} with the standard Euclidean metric Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x, y) = \mid x - y \mid} defined for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in M} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} is a metric space. Now, consider the following sequence in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (x_n)_{n=1}^{\infty} = \left ( 1 - \frac{1}{n} \right )_{n=1}^{\infty} = \left ( 0, \frac{1}{2}, \frac{2}{3}, ... \right ) \end{align}}

We claim that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_n)_{n=1}^{\infty}} is a Cauchy sequence. Let's prove this. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m, n \in \mathbb{N}} , and consider:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad d(x_m, x_n) = \left| \left ( 1 - \frac{1}{m} \right ) - \left ( 1 - \frac{1}{n} \right ) \right| = \left| \frac{1}{n} - \frac{1}{m} \right| \leq \left| \frac{1}{n} \right| + \left| \frac{1}{m} \right| = \frac{1}{n} + \frac{1}{m} \end{align}}

Choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N > \frac{2}{\epsilon}} . Then if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m, n \in \mathbb{N}} are such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m, n, \geq N} then:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad m \geq N > \frac{2}{\epsilon} \quad \mathrm{so} \quad \frac{1}{m} \leq \frac{1}{N} < \frac{\epsilon}{2} \\ \quad n \geq N > \frac{2}{\epsilon} \quad \mathrm{so} \quad \frac{1}{n} \leq \frac{1}{N} < \frac{\epsilon}{2} \end{align}}

Hence for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m, n \geq N} we have that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad d(x_m, x_n) \leq \frac{1}{n} + \frac{1}{m} < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align}}

So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_n)_{n=1}^{\infty}} is indeed a Cauchy sequence. However, it should be intuitively clear that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} x_n = 1} , but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \not \in [0, 1)} ! Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_n)_{n=1}^{\infty}} does not converge in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} and hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M, d)} is not a complete metric space.


Licensing

Content obtained and/or adapted from:

Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=Complete_Metric_Spaces&oldid=3654"