A commutative Banach algebra is a type of mathematical structure that combines the properties of a Banach space and an algebra. In this context, a Banach space is a complete normed vector space, meaning it has a way to measure the size of its elements and is complete in terms of limits. An algebra is a set equipped with operations like addition and multiplication that satisfy certain rules. The commutative aspect means that the multiplication operation is commutative, meaning that the order of multiplication does not affect the result. These algebras are important in functional analysis and have applications in various fields, including quantum mechanics and signal processing. They allow mathematicians to study functions and operators in a structured way, providing tools to analyze convergence and continuity. Examples of commutative Banach algebras include the algebra of continuous functions on a compact space and the algebra of bounded linear operators on a Hilbert space.