A Fréchet space is a type of topological vector space that is complete and metrizable. This means that it has a distance function that allows for the measurement of how close elements are to each other, and every Cauchy sequence in the space converges to a limit within the space. Fréchet spaces are often used in functional analysis and provide a framework for studying infinite-dimensional spaces. These spaces are defined by a translation-invariant metric that satisfies certain conditions, making them a generalization of both Banach spaces and Hilbert spaces. They are particularly useful in areas such as differential equations and the theory of distributions, where the properties of convergence and continuity are essential.