A Banach Space is a type of mathematical structure used in functional analysis, which is a branch of mathematics. It is defined as a complete normed vector space, meaning it has a way to measure the size of its elements (the norm) and every Cauchy sequence in the space converges to an element within the same space. This completeness property is crucial for many mathematical proofs and applications. In a Banach Space, the elements can be functions, sequences, or other mathematical objects, and the norm provides a way to quantify their distance from each other. Examples of Banach Spaces include the space of continuous functions and the space of p-integrable functions, denoted as L^p spaces. These spaces are fundamental in various areas of analysis and applied mathematics.