Normed Vector Space

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A Normed Vector Space is a vector space that is equipped with a norm function.



  • (Wikipedia, 2015) ⇒ Retrieved:2015-2-7.
    • In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial.

      1. The zero vector, '0, has zero length; every other vector has a positive length. :[math]\|x\|\gt 0[/math] if [math]x\ne0[/math]

      2. Multiplying a vector by a positive number changes its length without changing its direction. Moreover,
       :[math]\|\alpha x\|=|\alpha| \|x\|[/math] for any scalar [math]\alpha.[/math]

      3. The triangle inequality holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line. :[math]\|x+y\| \le \|x\|+\|y\|[/math] for any vectors x and y. (triangle inequality)

      The generalization of these three properties to more abstract vector spaces leads to the notion of norm. A vector space on which a norm is defined is then called a normed vector space.[1]

      Normed vector spaces are central to the study of linear algebra and functional analysis.

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