Suppose that $H$ is a Hilbert space and $M\subset H$ is a closed subset with non-empty interior and smooth boundary, whatever smooth boundary could mean. I wonder if the Gauss map on $\partial M$ is onto on the sphere, I hope yes.
The finite dimensional case, say dimension n, is easier (but not easy for me, I lack of enough training) and the problem can be stated better, using a n-dimensional manifold instead a closed subset, but I can not figure out what happens in $l^2$, is the hypothesis true or false?
The hypothesis holds obviously if $M$ is a closed ball.
We should assume that $M$ is bounded. For the finite dimensional case, take any unit vector $v$ and a plane with $v$ as a normal vector. Sweep the plane through the space until it touches $M$, the Gauss map at the first point of contact with $M$ is $v$. The problem in the infinite dimensional case is to verify the existence of the first point of contact. If $M$ is convex, that point exits.
To define smoothness I propose the following: suppose that $x\in\partial M$ and that there exists a functional $F$ defined in a neighborhood $U$ of $x$, such that $F=0$ on $\partial M\cap U$ and $dF(x)$ is a non-zero bounded linear functional. I don't know if I should define it differently.