Толық нормаланған векторлық кеңістіктер және Банах кеңістігі

Normed vector space that is complete
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces.

If and are normed spaces over the same ground field the set of all continuous linear maps is denoted by In infinite dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space to another normed space is continuous if and only if it is bounded on the closed unit ball of Thus, the vector space can be given the operator norm

For a Banach space, the space is a Banach space with respect to this norm. In categorical contexts, it is sometimes convenient to restrict the function space between two Banach spaces to only the short maps; in that case the space reappears as a natural bifunctor. If is a Banach space, the space forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps. If and are normed spaces, they are isomorphic normed spaces if there exists a linear bijection such that and its inverse are continuous. If one of the two spaces or is complete (or reflexive, separable, etc.) then so is the other space. Two normed spaces and are isometrically isomorphic if in addition, is an isometry, that is, for every in The Banach–Mazur distance between two isomorphic but not isometric spaces and gives a measure of how much the two spaces and differ.

A Banach algebra is a Banach space over or together with a structure of algebra over , such that the product map is continuous. An equivalent norm on can be found so that for all

When is separable, the unit ball of the dual is weak* compact by the Banach–Alaoglu theorem and metrizable for the weak* topology, If the dual is separable, the weak topology of the unit ball of is metrizable. This applies in particular to separable reflexive Banach spaces. Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences. A Banach space is reflexive if and only if each bounded sequence in has a weakly convergent subsequence. A weakly compact subset in is norm compact. Indeed, every sequence in has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of

A way to classify Banach spaces is through the probabilistic notion of type and cotype, these two measure how far a Banach space is from a Hilbert space.

If is an isometry from the Banach space onto the Banach space (where both and are vector spaces over ), then the Mazur–Ulam theorem states that must be an affine transformation. In particular, if this is maps the zero of to the zero of then must be linear. This result implies that the metric in Banach spaces, and more generally in normed spaces, completely captures their linear structure.

Finite dimensional Banach spaces are homeomorphic as topological spaces, if and only if they have the same dimension as real vector spaces. Anderson–Kadec theorem (1965–66) proves that any two infinite dimensional separable Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved that any two Banach spaces are homeomorphic if and only if they have the same density character, the minimum cardinality of a dense subset.

Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gateaux derivative for details. The Fréchet derivative allows for an extension of the concept of a total derivative to Banach spaces. The Gateaux derivative allows for an extension of a directional derivative to locally convex topological vector spaces. Fréchet differentiability is a stronger condition than Gateaux differentiability. The quasi derivative is another generalization of directional derivative that implies a stronger condition than Gateaux differentiability, but a weaker condition than Fréchet differentiability.

Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions or the space of all distributions on are complete but are not normed vector spaces and hence not Banach spaces. In Fréchet spaces one still has a complete metric, while LF spaces are complete uniform vector spaces arising as limits of Fréchet spaces.