Reproducing kernel Hilbert space - Wikipedia

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 converse does not need to be true. Informally, this can be shown by looking at the supremum norm: the sequence of functions sin 𝑛 ⁡ ( 𝑥 ) converges pointwise, but does not converge uniformly i.e. does not converge with respect to the supremum norm. (This is not a counterexample because the supremum norm does not arise from any inner product due to not satisfying the parallelogram law.) It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS.[1] Some examples, however, have been found.[2][3] L2 spaces are not Hilbert spaces of funct

Reproducing kernel Hilbert space - Wikipedia Jump to content From Wikipedia, the free encyclopedia In functional analysis, a Hilbert space Figure illustrates related but varying approaches to viewing RKHS In functional analysis , 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 L x : H → C {\display