By Lemmas 1 and 2 and

triangle inequality we derive

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

On the other hand, by the

triangle inequality, we can deduce

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