# Linear Operators in Hilbert Space - A Dense Question

## Main Question or Discussion Point

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?

mathman
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 ;)