Reproducing kernel Hilbert space

In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions f and g in the RKHS are close in norm, i.e., ‖f-g‖ is small, then f and g are also pointwise close, i.e., |f(x)-g(x)| is small for all x. The reverse need not be true. It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS. Note that L² spaces are not Hilbert spaces of functions (and hence not RKHSs), but rather Hilbert spaces of equivalence classes of functions (for example, the functions f and g defined by f(x)=0 and g(x)=1_{${\displaystyle \scriptstyle \mathbb {Q} }$} are equivalent in L²). However, there are RKHSs in which the norm is an L²-norm, such as the space of band-limited functions (see the example below). An RKHS is associated with a kernel that reproduces every function in the space in the sense that for any ${\displaystyle x}$ in the set on which the functions are defined, "evaluation at ${\displaystyle x}$" can be performed by taking an inner product with a function determined by the kernel. Such a reproducing kernel exists if and only if every evaluation functional is continuous.

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