A Banach space is a type of mathematical space that is complete and normed. This means it has a way to measure the size of its elements (the norm) and that every Cauchy sequence in the space converges to an element within the same space. Banach spaces are fundamental in functional analysis and are used to study various mathematical problems. Examples of Banach spaces include L^p spaces, which consist of functions whose p-th power is integrable, and ℝ^n, the n-dimensional real number space. These spaces provide a framework for understanding linear operators and other mathematical structures in a rigorous way.