Banach space theory is a branch of functional analysis that studies Banach spaces, which are complete normed vector spaces. A normed vector space is a set of vectors equipped with a function that measures the size or length of the vectors, while completeness means that every Cauchy sequence in the space converges to a limit within the space. This theory has important applications in various fields, including mathematics, physics, and engineering. It provides tools for solving linear equations, optimizing functions, and understanding the behavior of different mathematical structures, making it a fundamental area of study in modern analysis.