Complex sentence: 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.
Split into simpler sentences: 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.