# Linear Operators in Hilbert Space - A Dense Question

1. Mar 5, 2010

### e(ho0n3

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?

2. Mar 6, 2010

### 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.

3. Mar 6, 2010

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}.

4. Jul 12, 2010