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Difference between revisions of "Kernel of a linear operator"

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The linear subspace of the domain of definition of the 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|topological vector space]] is a closed linear subspace of this space. For locally convex spaces (cf. [[Locally convex space|Locally convex space]]), 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|adjoint operator]] has a weakly-dense range.
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The linear subspace of the [[domain of definition]] of a [[linear operator]] that consists of all vectors that are mapped to zero.  
  
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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.
  
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The ''nullity'' is the [[dimension]] of the kernel.
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====Comments====
 
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====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.L. Kelley,   I. Namioka,   "Linear topological spaces" , v. Nostrand (1963) pp. Chapt. 5, Sect. 21</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> J.L. Kelley, I. Namioka, "Linear topological spaces", v. Nostrand (1963) pp. Chapt. 5, Sect. 21</TD></TR>
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</table>

Latest revision as of 08:26, 28 April 2023

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
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=15024
This article was adapted from an original article by G.L. Litvinov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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