Discussion Overview
The discussion revolves around the properties of Hilbert spaces, particularly focusing on their dimensionality and separability, with specific reference to spaces defined over the quaternions. Participants explore definitions, distinctions between finite and infinite dimensional spaces, and the implications of these properties on separability.
Discussion Character
- Debate/contested
- Conceptual clarification
- Technical explanation
Main Points Raised
- One participant defines a Hilbert space as \(\mathbb{R}^n\) with the Euclidean norm, questioning its validity and the nature of spaces over the quaternions.
- Another participant notes that while finite dimensional Euclidean spaces and infinite dimensional Hilbert spaces are similar, they are typically treated as distinct concepts.
- There is uncertainty about the use of quaternions as a scalar field for vector spaces, with one participant suggesting it is possible but requires further exploration.
- A participant emphasizes that the definition of Hilbert space should focus on the presence of an inner product rather than dimensionality, and discusses the confusion surrounding the concept of dimension.
- It is mentioned that for a Hilbert space to be separable, it must have a countable dense subset, which is linked to the existence of a countable maximal orthonormal set.
- One participant references Bourbaki's definitions, arguing that dimensionality should not factor into the definition of a Hilbert space.
- Another participant cites Dieudonné's work, indicating a shared perspective on the definitions and properties of Hilbert spaces.
- There is a suggestion that the distinction between finite and infinite dimensional Hilbert spaces may not be of interest to many in the community.
- One participant expresses a desire to conclude the discussion, suggesting that the question may be irrelevant.
Areas of Agreement / Disagreement
Participants express differing views on the definitions and properties of Hilbert spaces, particularly regarding dimensionality and separability. There is no consensus on whether finite dimensional spaces should be classified as Hilbert spaces, and the relevance of the question itself is contested.
Contextual Notes
Some participants highlight the potential confusion surrounding the definitions of dimensionality and separability in Hilbert spaces, indicating that these concepts may depend on specific interpretations and definitions.