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This module includes a brief introduction to metric spaces at a suitable level of detail for study of signals and systems.


In may courses, concepts such as continuity and convergence are invoked without much discussion of their formal definitions, instead relying on the reader's intuitive understanding of these matters. However, for purposes of proofs, including some in the main body of material for this course, a greater rigor is required. This module will discuss metric spaces, a mathematical construct that provide a framework for the study continuity, convergence, and other related ideas in their most concrete but still formal senses. This is accomplished by formalizing a notion of the distance between two elements in a set. The intent in this and subsequent modules in this chapter is not to give a complete overview of the basic topics of analysis but to give a short introduction to those most important to discussion of signal processing in this course.

Metric spaces

A notion of distance

In many situations in signal processing it is often useful to have a concept of distance between the points in a set. This notion is mathematically formalized through the idea of a metric space. A metric space ( M , d ) is a set M together with a function d : M × M R that assigns distances between pairs of elements in M while satisfying three conditions. First, for every x , y M , d ( x , y ) 0 with d ( x , y ) = 0 if and only if x = y . Second, for every x , y M , d ( x , y ) = d ( y , x ) symmetrically. Third, for every x , y , z M , d ( x , y ) + d ( x , z ) d ( y , z ) , which is known as the triangle inequality.

There are, of course, several different possible choices of definitions for distances in a given set. Our typical intuitive understanding of distance in R n fits within this framework as the standard Euclidean metric

d ( x , y ) = | | x - y | | 2

as does the taxicab or Manhatten metric

d ( x , y ) = | | x - y | | 1

that sums individual components of vectors, representing, for example, distances traveled walking around city blocks. Another simple yet more exotic example is provided by the discrete metric on any set defined by

d ( x , y ) = 0 x = y 1 x y

in which all pairs of distinct points are equidistant from eachother but every point is distance zero from itself. One can check that these satisfy the conditions for metric spaces.

Relationship with norms

It is not surprising that norms, which provide a notion of size, and metrics, which provide a notion of distance, would have a close relationship. Intuitively, one way of defining the distance between two points in a metric space could be the size of their difference. In other words given a vector space V over the field F with norm | | · | | , we might ask if the function

d ( x , y ) = | | x - y | |

for every x , y V satisfies the conditions for ( V , d ) to be a metric space.

Let V be a vector space over the field F with norm | | · | | , and let d ( x , y ) = | | x - y | | . Recall that since | | · | | is a norm, | | x | | = 0 if and only if x = 0 and | | a x | | = | a | | | x | | for all a F and x V . Hence | | x - y | | 0 for all x , y V and | | x - y | | = 0 if and only if x = y . Since y - x = - ( x - y ) and | | - ( x - y ) | | = | | x - y | | it follows that | | x - y | | = | | y - x | | for all x , y V . Finally, | | x | | + | | y | | | | x + y | | by the properties of norms, so | | x - y | | + | | x - z | | | | y - z | | for all x , y , z V . Thus, ( V , d ) does indeed satisfy the conditions to be a metric space and is discussed as the metric space induced by the norm | | · | | .

Metric spaces summary

Metric spaces provide a notion of distance and a framework with which to formally study mathematical concepts such as continuity and convergence, and other related ideas. Many metrics can be chosen for a given set, and our most common notions of distance satisfy the conditions to be a metric. Any norm on a vector space induces a metric on that vector space and it is in these types of metric spaces that we are often most interested for study of signals and systems.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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