Mercer’s theorem is a fundamental result in functional analysis that provides conditions under which a symmetric positive semi-definite kernel can be represented as an inner product in a Hilbert space. This theorem is particularly useful in the study of integral operators and helps in understanding the properties of certain types of functions. The theorem states that if a continuous function defined on a compact space is symmetric and positive semi-definite, it can be expressed as a sum of eigenfunctions weighted by non-negative eigenvalues. This has applications in various fields, including machine learning, statistics, and signal processing, where it aids in dimensionality reduction and data analysis.