# Banach-Steinhaus theorem

## Banach-Steinhaus theorem

[¦bä‚näk ¦stīn‚hau̇s ‚thir·əm]
(mathematics)
If a sequence of bounded linear transformations of a Banach space is pointwise bounded, then it is uniformly bounded.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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and by Banach-Steinhaus theorem; it converges to zero uniformly on compact subsets of D, so we have
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.
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