SkipCaptcha
175
edits
Changes
Jump to navigation
Jump to search
Line 1:
Line 1: − + + + + + + + + +
norm is a non-mathematical article
A '''normed vector space''' is a [[vector space]] equipped with a length-measuring function called the [[norm]].
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 number]]s, 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.
[[Category:Linear algebra]]
[[Category:Linear algebra]]