Is the image of a dense set under an isometric operator is again dense set? i.e.,
Given two Hilbert spaces $X, Y$.
If $T:X\to Y$ is an isometric operator and $S$ is a dense span subset of $X$, is it true that $T(S)$ is a dense span subset of $T(X)$?.
If you are talking of an isometry onto $Y$ then it is true because any homeomorphism maps dense sets to dense sets. Otherwise it is certainly not true. Let $T: l^{2} \to l^{2}$ be defined by $T(x_1,x_2,...) =(0,x_1,x_2,...)$ and Take $S=X$. Then the closure of the range of $T$ is $\{(x_n):x_1=0\}$.