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