A Fréchet space is a type of topological vector space that is complete and metrizable. This means it has a distance function that allows for the measurement of convergence of sequences, and every Cauchy sequence in the space converges to a limit within the space. Fréchet spaces are often used in functional analysis and are particularly useful for studying spaces of functions. These spaces are defined by a translation-invariant metric that satisfies certain properties, such as being locally convex. Examples of Fréchet spaces include the space of continuous functions on a compact interval and the space of smooth functions on a manifold. They play a significant role in various areas of mathematics, including the study of differential equations and topology.