Linear Operators in Hilbert Space - A Dense Question

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In the context of Hilbert spaces, the discussion centers on whether a proper subset of linear operators can be dense in the set of all linear operators. It is established that the subset T, formed by linear combinations of basis elements with rational coefficients, is indeed dense in S, the set of linear operators. The discussion assumes that S is not a zero space and emphasizes the importance of the definition of a basis in this context. A trivial solution is mentioned, but it does not align with the conventional understanding of a basis in vector spaces. The conversation highlights the complexity and nuances involved in the properties of linear operators in Hilbert spaces.
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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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