# Kernel of a linear operator

From Encyclopedia of Mathematics

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.

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#### References

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

**How to Cite This Entry:**

Kernel of a linear operator.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Kernel_of_a_linear_operator&oldid=36011

This article was adapted from an original article by G.L. Litvinov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article