Difference between revisions of "Normed vector space"
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Every norm defines a metric in a canonical way: d(x, y) = |x - y|, but not every metric corresponds to a norm. Also, metrics define a topology in the space, and this make the space a topological vector space. | Every norm defines a metric in a canonical way: d(x, y) = |x - y|, but not every metric corresponds to a norm. Also, metrics define a topology in the space, and this make the space a topological vector space. | ||
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Latest revision as of 14:35, July 28, 2016
A normed vector space is a vector space equipped with a length-measuring function called the norm.
A norm is a function |.| taking as arguments vectors and returning real numbers, satisfying these properties:
- |a v| = |a| |v| for any scalar a and any vector v
- |v| > 0 unless v = 0
- |v + w| ≤ |v| + |w| for all vectors v and w (this is called the triangular inequality, because in a geometrical triangle every side is smaller than the sum of the other two)
Every norm defines a metric in a canonical way: d(x, y) = |x - y|, but not every metric corresponds to a norm. Also, metrics define a topology in the space, and this make the space a topological vector space.