"Almost" in the kernel Is there a notion of "almost kernel" of a matrix? Namely, a vector $v$ is "almost" in the kernel of the matrix if it is mapped "close" to 0. And thus the "almost kernel" of a matrix is a subspace (this would need to be verified) of all vectors that are mapped closed to zero. 
I guess one could define something like this, but I am wondering if such a notion already exists.

You can consider for fixed $\epsilon>0$ the set
$$
C:=\{v: \ \|Av\|\le \epsilon \|v\|\}.
$$
This has some nice properties: it is a closed and convex cone. Also $v\in C$ implies $-v\in C$. 
Such constructions are useful in constrained optimization and second-order sufficient optimality conditions. Second-order necessary optimality conditions are proven for directions in some critical cone. The critical cone has to be extended for sufficient conditions to work. There, constructions similar to the above are used. You can look in the book of Bonnans and Shapiro for the theory.
