Counterexamples of the Banach-Steinhaus Theorem in Functional Analysis

Functional analysis is a branch of mathematics that generalizes the notions of differentiation, integration, and vector spaces. One of its fundamental theorems is the Banach-Steinhaus theorem, which provides a convergence result for a family of linear operators. While there are numerous proofs affirming the validity of this theorem, exploring its conditions and hypotheses can lead us to discover scenarios where the theorem fails. This article delves into these counterexamples, particularly focusing on the relaxation of completeness in Banach spaces.

Understanding the Banach-Steinhaus Theorem

The Banach-Steinhaus theorem, also known as the uniform boundedness principle, states that if a family of continuous linear operators between Banach spaces is pointwise bounded, then it is uniformly bounded. This theorem is a cornerstone in functional analysis and has profound implications for the study of operator theory and functional spaces.

Dropping the Completeness Hypothesis

The most obvious hypothesis to relax is the completeness requirement of the Banach spaces involved. A Banach space is a complete normed vector space, meaning every Cauchy sequence converges. When completeness is dropped, the theorem no longer holds, as demonstrated by the following examples.

Counterexample with the Space ( c_{00} )

Consider the space ( c_{00} ), the space of finitely supported scalar sequences. This space is not a Banach space under any norm, as it lacks completeness. To see why the Banach-Steinhaus theorem fails in this case, we will construct a sequence of operators that demonstrates an unbounded behavior, even though these operators are pointwise bounded.

Define a family of linear maps ( T_n : c_{00} to c_{00} ) as follows:

For ( x x_1 x_2 ldots x_n 00 ldots ) in ( c_{00} ), let

( T_n x x_1 x_2 ldots x_{n-1} n x_n 00 ldots )

This operator shifts the last non-zero entry ( n ) to the end of the sequence, multiplying it by ( n ). For any ( x in c_{00} ), there exists an ( n ) large enough such that ( x T_n x ). However, the sequence ( T_n ) is unbounded because the norm of ( T_n x ) can be arbitrarily large. Therefore, even though each ( T_n ) is pointwise bounded, the whole family of operators is not uniformly bounded, violating the Banach-Steinhaus theorem.

Generalizing the Notion of Convergence

Another way to relax the hypotheses of the Banach-Steinhaus theorem is to modify the notion of convergence. Instead of using the standard norm convergence, we can consider filter convergence, a more general concept of convergence in topological spaces.

Filter convergence can be defined in terms of families of sets that capture the intuitive notion of a sequence getting arbitrarily close to some point. When using filter convergence instead of norm convergence, the theorem does not necessarily hold. Constructing counterexamples in this context involves showing that a pointwise bounded family of operators may not be uniformly bounded under filter convergence.

Conclusion

While the Banach-Steinhaus theorem is a powerful result in functional analysis, relaxing the conditions under which it holds can lead to interesting counterexamples. These counterexamples not only challenge our understanding of the theorem but also highlight the importance of the hypotheses in the statement of the theorem. By studying these cases, mathematicians can better appreciate the nuances and limitations of the theorem and deepen their understanding of the subject.