# triangle inequality

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## triangle inequality

[′trī‚aŋ·gəl ‚in·i′kwäl·ə·dē]
(mathematics)
For real or complex numbers or vectors in a normed space x and y, the absolute value or norm of x + y is less than or equal to the sum of the absolute values or norms of x and y.
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It follows from the definiton of [x.sub.n+1] and the triangle inequality that
By using the triangle inequality and Lemma 1, we can get the main conclusion for the [L.sup.2] norm error estimate of the UWDG scheme.
Using the triangle inequality and the fact that [M.sub.n](0,0) = n, we obtain
Also, Holder's inequality is used to prove Minkowski's inequality (the triangle inequality for [L.sub.p] spaces) and to establish that [L.sub.q]([mu]) is the dual space of [L.sub.p]([mu]) for p [member of] [1, [infinity]).
and from the triangle inequality for distances in (22a) and (22b) and the use of (18), one gets
By choosing [c.sub.1] = c [member of] [0, 2], noting that ([c.sub.1] + [alpha])([c.sub.1] + b) [greater than or equal to]([c.sub.1] - a)([c.sub.1] - b), where a,b [greater than or equal to] 0, applying the triangle inequality and replacing |y| by [mu] on the right-hand side of (3.7), we obtain
For f(z) given by (2) and using the triangle inequality we have
The topics include the triangle inequality, vectors and the dot product, and extremal points in triangles.
Besides, the distance defined in this paper satisfies the triangle inequality, which is another point different from those of previous related literatures.
When [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies the triangle inequality. According to Lemma 2.1, we have
Then, by using the triangle inequality and the fact that [absolute value of x] [less than or equal to] 1,we obtain
By using the triangle inequality and Cauchy-Schwarz inequality, we have

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