Kernel (mathematics)

The term kernel has two unrelated meanings in mathematics.

• In abstract algebra and related fields, the kernel of a homomorphism consists of all those elements which are sent to the identity element by the homomorphism; it is a structure which in many contexts describes how far a given homomorphism is from being injective. This concept is generalised in different ways in universal algebra and category theory. See kernel (algebra).
• In analysis, one considers integral operators which transform a function f into a function Tf given by (Tf)(x) = ∫ K(x,y)f(y) dy. The function K is called the kernel of the operator T. This usage applies also to convolution operators such as the Dirichlet kernel.