For ease of understanding , we provide the framework for real scalar - valued Hilbert spaces but also comment on the extension to Hilbert spaces over complex - valued functions as many important examples of the reproducing kernel Hilbert spaces are spaces of analytic functions .
For ease of understanding , we provide the framework for real - valued Hilbert spaces . 
 The theory can be easily extended to spaces of complex - valued functions and hence include the many important examples of the reproducing kernel Hilbert spaces that are spaces of analytic functions .