# Do Hilbert Space Isomorphism Map Dense Sets to Dense Sets?

1. Oct 1, 2013

### logarithmic

Suppose that H, K are Hilbert spaces, and A : H -> K is a bounded linear operator and an isomorphism.

If X is a dense set in H, then is A(X) a dense set in K?

Any references to texts would also be helpful.

2. Oct 1, 2013

### csopi

It is true. Suppose, that A(X) is not dense. Then let V be a non-empty open set in K \ A(X). The pullback of V by A is open (A is bounded) and not empty (A is isomorphism), and by definition it is not in X, which contradicts the fact, that X is dense in H.

Last edited: Oct 1, 2013