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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

so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
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what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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