and by

Banach-Steinhaus theorem; it converges to zero uniformly on compact subsets of D, so we have

Now by the

Banach-Steinhaus theorem, there is constant M such that

So by the

Banach-Steinhaus theorem, W is bounded in [L.sup.p] (0, [infinity]).

Thus, it follows by the Banach-Steinhaus theorem [24, p.

Hence, by the Banach-Steinhaus theorem there exists a [f.sub.1] [member of] P[W.sup.1.sub.[pi]] such that

By the Banach-Steinhaus theorem there exists a function hit such that

Then, by the Banach-Steinhaus theorem, we would have supn [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] < [infinity] which is not possible since, again by Proposition 2.3, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, by the Banach-Steinhaus theorem, we would have [sup.sub.n] [[parallel][[phi].sup.v.sub.n][parallel].sup.1/2.sub.4] < [infinity] which is not possible since, again by Proposition 2.4, [[parallel][[phi].sup.v.sub.n] [parallel].sup.1/2.sub.4] ~ [(log n).sup.1/8].

Uniform boundedness principle for bounded linear operators (

Banach-Steinhaus theorem) is one of the cornerstones of classical functional analysis (see, e.g., [1-3] and the references cited therein).

Applying the Banach-Steinhaus theorem completes the proof.

Then by the Banach-Steinhaus theorem there is a constant C12 such that

Hence, by the Banach-Steinhaus theorem there is a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that lim [sup.sub.N[right arrow][infinity]] [[parallel][[??].sub.N][f.sub.1][parallel].sub.[infinity]] = [infinity], which completes the proof.