Banachs

Banachs, more commonly called Banach spaces, are complete normed vector spaces that form a central object of study in functional analysis. By definition, a Banach space X is a vector space over the real or complex numbers equipped with a norm ||·|| such that X is complete with respect to the metric d(x,y) = ||x − y||. Completeness means every Cauchy sequence in X converges to a limit in X.

Typical examples include the sequence spaces l^p (1 ≤ p ≤ ∞) with the p-norm ||x||_p = (sum |x_n|^p)^{1/p} for p<∞ and the sup norm for p=∞; the space C(K) of continuous functions on a compact Hausdorff space K with the sup norm; and L^p spaces of p-integrable functions on a measure space, again with the p-norm (or essential sup norm for p=∞).

Banach spaces often possess additional structure. Many are separable, and a wide class—such as l^p and L^p for 1 < p < ∞—are reflexive and uniformly convex. These properties have implications for the geometry of the space and for convergence of sequences and operators.

Key theorems concern linear operators between Banach spaces: the Open Mapping Theorem, the Closed Graph Theorem, and the Uniform Boundedness Principle (Banach–Steinhaus). Together, they underpin much of functional analysis and its applications.

History and naming: Banachs are named after Stefan Banach, a Polish mathematician who helped found modern functional analysis. The concept emerged from his work in the early 20th century and became foundational through his and contemporaries’ development of operator theory and related results.

Applications span analysis, probability theory, partial differential equations, harmonic analysis, and mathematical physics, where Banach spaces provide a versatile framework for convergence, continuity, and approximation.