Reproducing kernel

In functional analysis, a Hilbert spaceFigure illustrates related but varying approaches to viewing RKHSIn functional analysis, a discipline within mathematics, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H{\displaystyle H} of functions from a set X{\displaystyle X} (to R{\displaystyle \mathbb {R} } or C{\displaystyle \mathbb {C} }) is an RKHS if the point-evaluation functional Lx:H→C{\displaystyle L_{x}:H\to \mathbb {C} }, Lx(f)=f(x){\displaystyle L_{x}(f)=f(x)}, is continuous for every x∈X{\displaystyle x\in X}. Equivalently, H{\displaystyle H} is an RKHS if for every x∈X{\textstyle x\in X} there exists a function Kx∈H{\displaystyle K_{x}\in H} such that, for all f∈H{\displaystyle f\in H},⟨f,Kx⟩=f(x).{\displaystyle \langle f,K_{x} angle =f(x).}The function Kx{\displaystyle K_{x}} is then called the reproducing kernel, and it reproduces the value of f{\displaystyle f} at x{\displaystyle x} via the inner product. An immediate consequence of this property is that convergence in norm implies uniform convergence on any subset of X{\displaystyle X} on w