A Banach algebra is a type of algebraic structure that combines the properties of a normed space and an algebra. It consists of a set of elements where you can perform addition and multiplication, and it also has a norm that allows you to measure the size of elements. The norm must satisfy certain conditions, making the algebra complete, meaning that every Cauchy sequence in the algebra converges to an element within the algebra. In a Banach algebra, the multiplication operation is associative, and there is a multiplicative identity element. These algebras are important in functional analysis and have applications in various fields, including quantum mechanics and signal processing. They help in studying linear operators and their properties in a rigorous mathematical framework.