A reproducing kernel is a special type of function used in functional analysis and machine learning. It allows for the evaluation of inner products in a Hilbert space, which is a complete vector space with an inner product. The key property of a reproducing kernel is that it reproduces the function values at specific points, meaning that for any function in the space, the value can be obtained by taking the inner product with the kernel function. In the context of support vector machines and Gaussian processes, reproducing kernels enable efficient learning and prediction. They provide a way to map input data into a higher-dimensional space, making it easier to find patterns and relationships. This concept is fundamental in kernel methods, which are widely used in various machine learning applications.