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What is the difference between norm and modulus?

  1. Aug 27, 2007 #1

    cks

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    norm is defined to be the length of the vector and we put we denote it by ||a||.

    However, modulus |a| also means the length of a from the origin?

    So, what is the difference between the symbol || || and | |?
     
  2. jcsd
  3. Aug 27, 2007 #2

    quasar987

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    There is none. Some authors use one, others use the other.
     
  4. Aug 27, 2007 #3
    I was always taught that the norm is defined to be any length which satisfies the definition of a norm, but modulus is specifically the Euclidean norm. That could have just been the instructor/book, though I suppose.
     
  5. Aug 27, 2007 #4

    dextercioby

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    The modulus is the norm on [itex]\mathbb{R} [/itex]. [itex] \left(\mathbb{R},\left| , \right| \right) [/itex] is a Banach space.
     
  6. Aug 31, 2007 #5

    cks

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    I think norm is the length of a point from the origin.

    whereas the modulus is more of a distance from one point to another point.

    norm is just a specific case of the distance from a point to its origin.
     
  7. Aug 31, 2007 #6

    D H

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    The semi-standard usage is that modulus is specialized to the reals (absolute value), complex numbers (complex modulus), and quaternions. However, some write [itex]|x|[/itex] instead of [itex]||x||[/itex] to mean norm. Norm is a very generalized concept that covers everything from Euclidean distance to distance as measured on a road grid (taxicab norm) to the L-infinity norm, and beyond. Anything that qualifies as a distance can be used as a norm. Even things much more complex and abstract than vectors can have a norm. Well-defined norms exist for matrices, for example.
     
  8. Aug 31, 2007 #7

    Chris Hillman

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    Actually, both norm and modulus are potentially ambiguous terms in mathematics; the intended meaning depends upon context. Generally speaking, modulus often suggests algebraico-geometric-analytic origins (as in the theory of elliptic functions) while norm tends to suggest operator algebras and functional analysis. In the context of functional analysis, esp. Banach spaces, I agree with DH about what one can reasonably expect "modulus" to denote.
     
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