Editing Triangle inequality (section)

==Normed vector space==
[[File:Vector-triangle-inequality.svg|thumb|300px|Triangle inequality for norms of vectors.]]
In a [[normed vector space]] {{mvar|V}}, one of the defining properties of the [[norm (mathematics)|norm]] is the triangle inequality:

:<math> \|\mathbf u + \mathbf v\| \leq \|\mathbf u\| + \|\mathbf v\| \quad \forall \, \mathbf u, \mathbf v \in V</math>

That is, the norm of the [[Vector sum#Addition and subtraction|sum of two vectors]] is at most as large as the sum of the norms of the two vectors.  This is also referred to as [[subadditivity]]. For any proposed function to behave as a norm, it must satisfy this requirement.<ref name=Kress>
{{cite book |title=Numerical analysis |author=Rainer Kress |chapter=§3.1: Normed spaces |chapter-url=https://books.google.com/books?id=e7ZmHRIxum0C&pg=PA26 |page=26 |isbn=0-387-98408-9 |year=1988 |publisher=Springer}}</ref>

If the normed space is [[euclidean space|Euclidean]], or, more generally, [[strictly convex space|strictly convex]], then <math>\|\mathbf u+\mathbf v\|=\|\mathbf u\|+\|\mathbf v\|</math> if and only if the triangle formed by {{math|'''u'''}}, {{math|'''v'''}}, and {{math|'''u''' + '''v'''}}, is degenerate, that is, {{math|'''u'''}} and {{math|'''v'''}} are on the same ray, i.e., {{math|'''u''' {{=}} 0}} or {{math|'''v''' {{=}} 0}}, or {{math|'''u''' {{=}} ''α'' '''v'''}} for some {{math|''α'' > 0}}. This property characterizes strictly convex normed spaces such as the {{math|''ℓ<sub>p</sub>''}} spaces with {{math|1 < ''p'' < ∞}}. However, there are normed spaces in which this is not true. For instance, consider the plane with the {{math|''ℓ''<sub>1</sub>}} norm (the [[Manhattan distance]]) and denote {{math|'''u''' {{=}} (1, 0)}} and {{math|'''v''' {{=}} (0, 1)}}. Then the triangle formed by {{math|'''u'''}}, {{math|'''v'''}}, and {{math|'''u''' + '''v'''}}, is non-degenerate but

:<math>\|\mathbf u+\mathbf v\|=\|(1,1)\|=|1|+|1|=2=\|\mathbf u\|+\|\mathbf v\|.</math>

===Example norms===
The ''[[absolute value]]'' is a norm for the [[real line]]; as required, the absolute value satisfies the triangle inequality for any real numbers {{mvar|u}} and {{mvar|v}}. If {{mvar|u}} and {{mvar|v}} have the same sign or either of them is zero, then <math>|u + v| = |u|+|v|.</math> If {{mvar|u}} and {{mvar|v}} have opposite signs, then [[without loss of generality]] assume <math>|u| > |v|.</math> Then {{nobr|<math>|u + v| = |u| - |v| < |u| + |v|.</math> Combining these cases:<ref name=Stewart>A proof not requiring separate cases is as follows: Any number is always less than or equal to its own absolute value, so <math>-|u| \leq u \leq |u|</math> and <math>-|v| \leq v \leq |v|.</math> Adding these inequalities together, <math>-\bigl(|u| + |v|\bigr) \leq u+v \leq |u| + |v|,</math> the sum has the form <math>-a \leq b \leq a</math> for <math>a = |u| + |v|</math> and <math>b = u + v.</math> But this always implies <math>| b | \leq a</math>, or, expanded, <math>|u + v| \leq |u|+|v|.</math> {{pb}} {{cite book |page=A10 |author=James Stewart |title=Essential Calculus |url=https://archive.org/details/studentsolutions0000stew |url-access=registration |isbn=978-0-495-10860-3 |publisher=Thomson Brooks/Cole |year=2008}}</ref>}}

<math display="block">|u + v| \leq |u|+|v|.</math>

The triangle inequality is useful in [[mathematical analysis]] for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers. There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers {{mvar|u}} and {{mvar|v}}, <math>|u-v| \geq \bigl||u|-|v|\bigr|.</math>

The ''[[taxicab norm]]'' or 1-norm is one generalization absolute value to higher dimensions. To find the norm of a vector <math>v = (v_1, v_2, \ldots v_n ),</math> just add the absolute value of each component separately, <math display="block">\|v\|_1 = |v_1| + |v_2| + \dotsb + |v_n|.</math>

The ''Euclidean norm'' or 2-norm defines the length of translation vectors in an {{mvar|n}}-dimensional [[Euclidean space]] in terms of a [[Cartesian coordinate system]]. For a vector <math>v = (v_1, v_2, \ldots v_n ),</math> its length is defined using the {{mvar|n}}-dimensional [[Pythagorean theorem]]: <math display="block">\|v\|_2 = \sqrt{|v_1|^2 + |v_2|^2 + \dotsb + |v_n|^2}.</math>

The ''inner product'' is norm in any [[inner product space]], a generalization of Euclidean vector spaces including infinite-dimensional examples. The triangle inequality follows from the [[Cauchy–Schwarz inequality]] as follows: Given vectors <math>u</math> and <math>v</math>, and denoting the inner product as <math>\langle u , v\rangle </math>:<ref name= Stillwell>{{cite book |title=The four pillars of geometry |author=John Stillwell |page=[https://archive.org/details/fourpillarsofgeo0000stil/page/80 80] |url=https://archive.org/details/fourpillarsofgeo0000stil |url-access=registration |isbn=0-387-25530-3 |year=2005 |publisher=Springer}}</ref>
:{|
|<math>\|u + v\|^2</math> || <math>= \langle u + v, u + v \rangle</math>
|-
| || <math>= \|u\|^2 + \langle u, v \rangle + \langle v, u \rangle + \|v\|^2</math>
|-
| || <math>\le \|u\|^2 + 2|\langle u, v \rangle| + \|v\|^2</math>
|-
| || <math>\le \|u\|^2 + 2\|u\|\|v\| + \|v\|^2</math> (by the Cauchy–Schwarz inequality)
|-
| || <math>=  \left(\|u\| + \|v\|\right)^2</math>.
|}

The Cauchy–Schwarz inequality turns into an equality if and only if {{mvar|u}} and {{mvar|v}} are linearly dependent. The inequality <math>\langle u, v \rangle + \langle v, u \rangle \le 2\left|\left\langle u, v \right\rangle\right| </math> turns into an equality for linearly dependent <math>u</math> and  <math>v</math> if and only if one of the vectors {{mvar|u}} or {{mvar|v}} is a ''nonnegative'' scalar of the other. Taking the square root of the final result gives the triangle inequality.

The [[p-norm|{{mvar|p}}-norm]] is a generalization of taxicab and Euclidean norms, using an arbitrary positive integer exponent, <math display="block">\|v\|_p = \bigl(|v_1|^p + |v_2|^p + \dotsb + |v_n|^p\bigr)^{1/p}, </math> where the {{math|''v<sub>i</sub>''}} are the components of vector {{mvar|v}}.

Except for the case {{math|1=''p'' = 2}}, the {{mvar|p}}-norm is ''not'' an inner product norm, because it does not satisfy the [[parallelogram law]]. The triangle inequality for general values of {{mvar|p}} is called [[Minkowski's inequality]].<ref name=Saxe>{{cite book |title=Beginning functional analysis |author= Karen Saxe|author-link= Karen Saxe |url=https://books.google.com/books?id=0LeWJ74j8GQC&pg=PA61 |page=61 |isbn=0-387-95224-1 |publisher=Springer |year=2002}}</ref> It takes the form:<math display="block">\|u+v\|_p \le \|u\|_p + \|v\|_p \ .</math>