By Lemmas 1 and 2 and triangle inequality
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
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