Page 250 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 22: The Uniform Boundedness Theorem

Conversely, let us assume that f o T is continuous for each f N*. To show that T is continuous Notes
it suffices to show that
T(B) = {Tx : x N, B = x 1} is bounded in N .

For each f N*, f o T is continuous and linear on N and so (f o T) B = f (T(B)) is bounded set for
every f N*, where we have considered the unit sphere B with centre at the origin and radius 1.
Since any bounded set in N can be obtained from B, T (B) is bounded by a non-empty subset X of
a normed linear space N of bounded f (X) is a bounded set of number for each f in N*.

22.2 Summary

 Uniform Boundedness Theorem: If (a) B is Banach space and N a normed linear space,
(b) {T } is non-empty set of continuous linear transformation of B into N and (c) {T (x)} is a
i i
bounded subset of N for each x B, then { T } is a bounded set of numbers, i.e. {T } is
i i
bounded as a subset of (B, N).
 If B is a Banach space and (f (x)) is sequence of continuous linear functionals on B such that
i
(|f (x)|) is bounded for every x B, then the sequence ( T ) is bounded.
i i
22.3 Keywords

Imbedding: Imbedding is one instance of some mathematical structure contained within another
instance, such as a group that is a subgroup.

Uniform Boundedness Theorem: The uniform boundedness theorem, like the open mapping
theorem and the closed graph theorem, is one of the cornerstones of functional analysis with
many applications.

22.4 Review Questions

1. If X is a Banach space and A X*, then prove that A is a bounded set if and only if for every
x in X, Sup {|f (x)| : f A} < .
2. Let  be a Hilbert space and let  be an orthonormal basis for . Show that a sequence {h }
n
in satisfies <h , h> 0 for every h in if and only if sup { h : n 1} < and <h , e>
n n n
0 for every e in .
22.5 Further Readings

Books Bourbaki, Nicolas, Topological vector spaces, Elements of mathematics, Springer (1987).
Diendonne, Jean, Treatise on Analysis, Volume 2, Academic Press, (1970).
Rudin, Walter, Real and Complex Analysis, McGraw-Hill, 1966.

Online links www.jstor.org/stable/2035429
www.sciencedirect.com/science/article/pii/S0168007211002004

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