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How does a norm differ from an absolute value?

  1. May 18, 2006 #1
    How does a norm differ from an absolute value? For example, is

    [tex]
    \|\mathbf{x}\| = \sqrt{x_1^2 + \cdots + x_n^2}
    [/tex]

    any different than

    [tex]
    |\mathbf{x}| = \sqrt{x_1^2 + \cdots + x_n^2}
    [/tex]

    ??
     
  2. jcsd
  3. May 18, 2006 #2

    shmoe

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    The usual absolute value satisfies all the properties of a norm- it's just the most common example of one. You've surely met other examples of norms though.
     
  4. May 18, 2006 #3
    What the norm is kind of depends on how you are defining the inner product. The example you have is for a normal dot product, but the norm for an inner product is the sqrt of the inner product....

    A better definition
    http://mathworld.wolfram.com/VectorNorm.html
     
    Last edited: May 18, 2006
  5. May 18, 2006 #4

    shmoe

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    Every inner product leads to a norm, but not all norms come from an inner product.
     
  6. May 19, 2006 #5

    HallsofIvy

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    Strictly speaking, the term "absolute value" is used only for numbers: real or complex. The term "norm" is used for vectors. The norm is exactly the same as what you are calling absolute value but I wouldn't use that term.
     
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