In mathematics, more specifically in functional analysis, a Banach space (/ ˈbɑː.nʌx /, Polish pronunciation: [ˈba.nax]) is a complete normed vector space.

A normed vector space V is called a Banach space if every Cauchy sequence in V converges. That is, a Banach space is a complete normed vector space.

Apr 13, 2019 · Banach spaces were named after S. Banach who in 1922 began a systematic study of these spaces [Ba], based on axioms introduced by himself, and who obtained highly …

Dec 3, 2025 · While a Hilbert space is always a Banach space, the converse need not hold. Therefore, it is possible for a Banach space not to have a norm given by an inner product.

Since any two norms on a finite dimensional space are equivalent, by Proposition 1.4, it follows that any finite dimensional normed vector space is a Banach space.

5.1 Banach spaces A normed linear space is a metric space with respect to the metric d derived from its norm, where d(x; y) = kx yk. De nition 5.1 A Banach space is a normed linear space …

Aug 19, 2025 · In mathematics, particularly in functional analysis, we work with a type of space called a Banach space. This is a collection of objects (often functions or sequences) that can …

Jul 1, 2025 · Thus a Banach space may equivalently be defined as a vector space equipped with a complete homogeneous translation-invariant metric. Actually, one usually sees a sort of …

Lecture 1: Basic Banach Space Theory Description: An introduction to Banach space theory, including vector spaces, norms and important examples of normed spaces.

Definition 1.2 A Banach space is a normed vector space such that is complete under the induced by the norm .