Kernel (mathematics)

The term kernel has several meanings in mathematics, some related to each other, some unrelated.

In analysis, one consider an integral operator T which transforms a function f into a function Tf given by the integral formula

The function K that appears in this formula is called the kernel of the operator T. This usage applies also to convolution operators such as the Dirichlet kernel.

Unrelated to this, if f is any function in any context, then the kernel of f is a certain equivalence relation on the domain of f which is defined in terms of f. For more on this in general, see Kernel (function).

This notion is used heavily in abstract algebra. But in the case of Mal'cev algebras, it can be replaced by a simpler definition; the kernel of a homomorphism f is the preimage under f of the zero element of the codomain. For more on this, see Kernel (algebra).

Finally, for this last notion of kernel is generalised in a certain sense in category theory; the kernel of a morphism f is the difference kernel of f and the corresponding zero morphism (if this exists). For more on this, see Kernel (category theory).