Difference between revisions of "Kernel of a linear operator"
From Encyclopedia of Mathematics
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| − | The linear subspace of the domain of definition of | + | 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 space]]s, 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. | ||
====Comments==== | ====Comments==== | ||
Revision as of 19:25, 31 December 2014
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.
Comments
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
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