Since the right-hand side converges to zero as n -> a, we obtain y € R . But since y s Y and II y 11 = 1, this is a contradiction.

Here are four examples to show that each of the four assumptions is indispensable to guarantee the boundedness. Note that each example violates only one of the four conditions.

The first example shows that assumption (i) is indispensable:

Fisure 16AA2A

Tne second example shows that (ii) is indispensable:

Fisure 16AA.2.2

1 fii!) is indispensable.

Tne third examples snows uu; 13

Figure 16.AA..2.3

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