Banach-Steinhaus theorem
Introduction
The Banach-Steinhaus theorem, also known as the Uniform Boundedness Principle, is a fundamental theorem in the field of Functional analysis. This theorem is named after mathematicians Stefan Banach and Hugo Steinhaus. The theorem is a significant result in functional analysis, having applications in various areas of mathematics, such as partial differential equation and Fourier analysis.
Statement of the Theorem
The Banach-Steinhaus theorem is stated as follows:
Let X be a Banach space and Y be a Normed space. If T_n : X → Y is a sequence of linear maps such that for every x in X, the sequence T_n x is bounded in Y, then there exists a constant C such that for all n and all x in X, we have ||T_n x|| ≤ C ||x||.
This theorem essentially states that if a sequence of linear maps is pointwise bounded, then it is uniformly bounded.
Proof of the Theorem
The proof of the Banach-Steinhaus theorem is based on the Baire category theorem. The proof is a classic example of a technique known as the "Baire category method", which involves the use of topological properties of complete metric spaces and the Baire category theorem.
The proof proceeds by contradiction. Assume that the sequence of linear maps is not uniformly bounded. This means that for each n, there exists an x_n in X such that ||T_n x_n|| > n ||x_n||. The set of all x in X such that ||T_n x|| ≤ n ||x|| is then closed and its complement in X is of the first category by the Baire category theorem. This leads to a contradiction, proving the theorem.
Applications of the Theorem
The Banach-Steinhaus theorem has wide-ranging applications in various areas of mathematics.
In the field of partial differential equations, the theorem is used to prove the existence of solutions. The theorem ensures that a sequence of approximations to a solution is uniformly bounded, which then allows the use of the Bolzano–Weierstrass theorem to extract a convergent subsequence.
In Fourier analysis, the theorem is used to prove the Riemann-Lebesgue lemma, which states that the Fourier transform of an integrable function tends to zero at infinity.