Abstract
The aim of this chapter is to explore the ramifications of our general metrization theory for classic functional analysis concerned with open mapping theorems, closed graph theorems, and uniform boundedness principles, for which we establish a new generation of results. Here we also prove a refinement of the classic Birkhoff–Kakutani theorem by fully bringing the topology into focus. The prevalent setting in most of the results established in this chapter is that of quasi-pseudonormed group.
Keywords
- Uniform Boundedness Principle (UBP)
- General Metrization Theory
- Open Mapping Theorem (OMT)
- Closed Graph Theorem (CGT)
- Modern Functional Analysis
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Notes
- 1.
In contrast to the more common practice, here we do not make the background assumption that the group in question is Abelian.
- 2.
Formula (6.110) is equivalent to \(\psi (y {_\ast} x) = \psi (x {_\ast} y)\) for all \(x,y \in G\), a condition also referred to as Abelian norm property.
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Mitrea, D., Mitrea, I., Mitrea, M., Monniaux, S. (2013). Functional Analysis on Quasi-Pseudonormed Groups. In: Groupoid Metrization Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8397-9_6
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