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While vector spaces have additional structure compared to a metric space, a general vector space has no notion of “length” or “distance.”

Definition 1

Let V be a vector space over K . A norm is a function · : V R such that

  • x 0 x V
  • x = 0 iff x = 0
  • α x = | α | x x V , α K
  • x + y x + y x , y V

A vector space together with a norm is called a normed vector space (or normed linear space ).

  • V = R N : x 2 = i = 1 N | x i | 2
    An illustration showing a point x in R2 and it's ell_2 (Euclidian) norm.  The norm is equal to the length of a straight line connecting x to the origin.
  • V = R N : x 1 = i = 1 N | x i | (“Taxicab”/“Manhattan” norm)
  • V = R N : x = max i = 1 , . . . , N | x i |
    An illustration showing a point x in R2 and it's ell_infinity norm.  The norm is equal to the length of the longer of the two (orthogonal) paths that connect x to the x- and y-axes.
  • V = L p [ a , b ] , p [ 1 , ) : x ( t ) p = a b | x ( t ) | p d t 1 / p (The notation L p [ a , b ] denotes the set of all functions defined on the interval [ a , b ] such that this norm exists, i.e., x ( t ) p < .)

Note that any normed vector space is a metric space with induced metric d ( x , y ) = x - y . (This follows since x - y = x - z + z - y x - z + y - z .) While a normed vector space “feels like” a metric space, it is important to remember that it actually satisfies a great deal of additional structure.

Technical Note: In a normed vector space we must have (from N2) that x = y if x - y = 0 . This can lead to a curious phenomenon when dealing with continuous-time functions. For example, in L 2 ( [ a , b ] ) , we can consider a pair of functions like x ( t ) and y ( t ) illustrated below. These functions differ only at a single point, and thus x ( t ) - y ( t ) 2 = 0 (since a single point cannot contribute anything to the value of the integral.) Thus, in order for our norm to be consistent with the axioms of a norm, we must say that x = y whenever x ( t ) and y ( t ) differ only on a set of measure zero. To reiterate x = y x ( t ) = y ( t ) t [ a , b ] , i.e., when we treat functions as vectors, we will not interpret x = y as pointwise equality, but rather as equality almost everywhere .

A smooth function defined on the interval [-1,1]. A function that is identical to the previous function, except for a point discontinuity where it takes a different value.

Questions & Answers

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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Digital signal processing. OpenStax CNX. Dec 16, 2011 Download for free at http://cnx.org/content/col11172/1.4
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