Linear Operators in Hilbert Space - A Dense Question

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Discussion Overview

The discussion revolves around the properties of linear operators in a Hilbert space, specifically whether there exists a proper subset of the set of linear operators that is dense in the set of all linear operators. The scope includes theoretical considerations and mathematical reasoning related to vector spaces and bases.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant posits that there exists a proper subset of linear operators that is dense in the set of all linear operators on a Hilbert space.
  • Another participant suggests defining a subset of linear operators using linear combinations of basis elements with rational coefficients, claiming this subset is dense in the larger set.
  • A third participant notes that the original question assumes the Hilbert space is non-trivial (not a zero space).
  • One participant challenges the definition of "basis" used in the discussion, indicating that the problem may be trivial if the usual definition of a basis is applied.
  • A later reply comments on the historical context of the question, implying it has been discussed previously.

Areas of Agreement / Disagreement

Participants express differing views on the definition of a basis and the implications of the density of subsets of linear operators. The discussion remains unresolved regarding the nature of the subset and the validity of the claims made.

Contextual Notes

The discussion does not clarify the assumptions regarding the properties of the Hilbert space or the completeness of the definitions used for bases and density.

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Let H be a Hilbert space and let S be the set of linear operators on H. Is there a proper subset of S that is dense in S?
 
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S is a vector space, so it must have a basis. Let T be a subset of S defined by linear combinations of basis elements with rational coefficients. T is a proper subset of S and is dense in S.
 
The question assumes that H (and thus S) is not a zero space, of course.

A more trivial solution would be to consider S \ {0}.
 
Not if his meaning of "basis" refers to the usual definition of the basis of a vector space. (That is, B is a basis of S iff every element of S is uniquely expressed as a finite linear combination of elements of B.) Of course, as stated, the problem is rather trivial.

Way to bring up an old thread ;)
 

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