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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From Claim 2 and the triangle inequality, we also have [parallel][y.sup.i.sub.m] - [z.sup.i.sub.m][parallel] [right arrow] 0.
It follows from the definiton of [x.sub.n+1] and the triangle inequality that
Using the triangle inequality and the fact that [M.sub.n](0,0) = n, we obtain
and from the triangle inequality for distances in (22a) and (22b) and the use of (18), one gets
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]).
Since [d.sub.v,a,b,c][member of] A, then, by the Schwarz Lemma, triangle inequality, and (29), we obtain
The topics include the triangle inequality, vectors and the dot product, and extremal points in triangles.
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
To estimate the second term above, we utilize a triangle inequality and (2.2), giving
It follows from the triangle inequality, the trace inequality, and the Poincare-Friedrichs inequality that

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