Reproducing kernel Hilbert space

（重定向自Reproducing kernel）

Figure illustrates related but varying approaches to viewing RKHS

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 L 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_{$\scriptstyle\mathbb{Q}$} are equivalent in L). However, there are RKHSs in which the norm is an L-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 $x$ in the set on which the functions are defined, "evaluation at $x$ " 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.

单词	Reproducing kernel
释义	Reproducing kernel 英语百科 Reproducing kernel Hilbert space （重定向自Reproducing kernel） 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 L 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_{$\scriptstyle\mathbb{Q}$} are equivalent in L). However, there are RKHSs in which the norm is an L-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 $x$ in the set on which the functions are defined, "evaluation at $x$ " 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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