Content deleted Content added
m slightly improved handling of coefficients in calculation in example |
Equation was broken |
||
Line 25:
{{NumBlk|:|<math> |L_x(f)| := |f(x)| \le M_x\, \|f\|_H \qquad \forall f \in H. \,</math>|{{EquationRef|1}}}}
Although <math>M_x<\infty</math> is assumed for all <math>x \in X</math>, it might still be the case that <math display="inline">\sup_x M_x = \infty</math>.
While property ({{EquationNote|1}}) is the weakest condition that ensures both the existence of an inner product and the evaluation of every function in <math>H</math> at every point in the ___domain, it does not lend itself to easy application in practice. A more intuitive definition of the RKHS can be obtained by observing that this property guarantees that the evaluation functional can be represented by taking the inner product of <math> f </math> with a function <math> K_x </math> in <math>H</math>. This function is the so-called '''reproducing kernel'''{{Citation needed|date=September 2022}} for the Hilbert space <math>H</math> from which the RKHS takes its name. More formally, the [[Riesz representation theorem]] implies that for all <math>x</math> in <math>X</math> there exists a unique element <math> K_x </math> of <math>H</math> with the reproducing property,
| |||