设X{\Bbb X}X是输入空间(欧氏空间RnR^nRn的子集或离散集合),又设H{\Bbb H}H为特征空间(希尔伯特空间),如果存在从x到H{\Bbb H}H的映射ϕ(x):X→H \phi(x): {\Bbb X} \rightarrow {\Bbb H} ϕ(x):X→H 使得对所有x,z∈Xx,z \in {\Bbb X}x,z∈X,函数K(x,z)K(x,z)K(x,z)满足条件:K(x,z)=ϕ(x)⋅ϕ(z) K(x, z) = \phi(x) \cdot \phi(z) K(x,z)=ϕ(x)⋅ϕ(z) 则称K(x,z)K(x,z)K(x,z)为核函数,ϕ(x)\phi(x)ϕ(x)为映射函数,式中ϕ(x)⋅ϕ(z)\phi(x) \cdot \phi(z)ϕ(x)⋅ϕ(z)为ϕ(x)\phi(x)ϕ(x)和ϕ(z)\phi(z)ϕ(z)的内积。
在学习与预测中只定义核函数K(x,z)K(x,z)K(x,z),而不显式地定义映射函数ϕ\phiϕ。