Abstract
In this chapter we present the open mapping theorem, the inverse operator theorem, the closed graph theorem, and the uniform boundedness principle. All these results belong to the circle of classical “fundamental principles of functional analysis” and have multiple applications. Some of such applications are given in this chapter, in particular applications to complementability of subspaces, to boundedness of partial sums operators with respect to a Schauder basis, and to Fourier series on an interval.
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Notes
- 1.
To Juliusz Schauder, a prominent member of the Lviv school of mathematics, we owe many fruitful ideas. For example, Schauder was the first to use Baire’s theorem to prove the open mapping theorem; to him also belongs the theorem on the compactness of the adjoint operator (Theorem 3 in Subsection 11.3.2), which lies at the foundations of the theory of compact operators. Also, it is clearly hard to overestimate the importance of the fixed-point principle (Subsection 15.1.4) and of the concept of a Schauder basis (Subsection 10.5). Thus, whenever hearing Schauder’s name, the reader should be ready to grasp something valuable for his mathematical culture.
- 2.
By definition, a subspace of a Banach space is a closed linear subspace, so the word “closed” in the statement of this theorem is superfluous. We emphasize the closedness here because in the previous chain of results we were speaking about subspaces of linear spaces, which were just linear subspaces. Also, for a continuous projector P in a normed space X its image P(X) is automatically closed, because it is equal to \(\mathrm{Ker}(I-P)\).
- 3.
The usage of the notation \(C(\mathbb {T})\) is explained in Exercise 2 below.
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Kadets, V. (2018). Classical Theorems on Continuous Operators. In: A Course in Functional Analysis and Measure Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-92004-7_10
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DOI: https://doi.org/10.1007/978-3-319-92004-7_10
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