Kernel of a linear operator

The linear subspace of the domain of definition of a linear operator that consists of all vectors that are mapped to zero.

The kernel of a continuous linear operator that is defined on a topological vector space is a closed linear subspace of this space. For locally convex spaces, a continuous linear operator has a null kernel (that is, it is a one-to-one mapping of the domain onto the range) if and only if the adjoint operator has a weakly-dense range.

The nullity is the dimension of the kernel.

References

[a1]	J.L. Kelley, I. Namioka, "Linear topological spaces", v. Nostrand (1963) pp. Chapt. 5, Sect. 21