A complementary triangle inequality in Hilbert and Banach spaces

or else yp = 0 and Zi = • • ■ = z„. In the course of his proof of inequality (2), Wilf derives as an intermediate auxiliary inequality the following: (3) (costfOd zi| + ■ • • + | s.| ) = | f,+ • • • + ft,| . Since inequality (2) follows readily from (3) by an application of the arithmetic-geometric mean inequality for real numbers, it is clear that inequality (3) plays the more fundamental r61e. In fact, inequality (3) may be interpreted as a triangle inequality, i.e., an inequality which runs the other way from the usual triangle inequality. The complementary character of (3), relative to the usual triangle inequality may be described as follows. The usual triangle inequality states that, for any complex zi, • • • , zn, one has 0-(| «j | + • • •+ | ft, |) ^ |zi+ • • -+z„| gl-(|zi| + •••+ k|).