In mathematics, more specifically in functional analysis, a Banach space (, ) 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.
Definition
A Banach space is a complete normed space A normed space is a pairIt is common to read instead of the more technically correct but (usually) pedantic especially if the norm is well known (for example, such as with spaces) or when there is no particular need to choose any one (equivalent) norm over any other (especially in the more abstract theory of topological vector spaces), in which case this norm (if needed) is often automatically assumed to be denoted by However, in situations where emphasis is placed on the norm, it is common to see written instead of The technically correct definition of normed spaces as pairs may also become important in the context of category theory where the distinction between the categories of normed spaces, normable spaces, metric spaces, TVSs, topological spaces, etc. is usually important. consisting of a vector space over a scalar field (where is commonly or ) together with a distinguishedThis means that if the norm is replaced with a different norm on then is the same normed space as not even if the norms are equivalent. However, equivalence of norms on a given vector space does form an equivalence relation. norm Like all norms, this norm induces a translation invariantA metric on a vector space is said to be translation invariant if for all vectors This happens if and only if for all vectors A metric that is induced by a norm is always translation invariant. distance function, called the canonical or (norm) induced metric, defined for all vectors byBecause for all it is always true that for all So the order of and in this definition does not matter.
This makes into a metric space A sequence is called or or if for every real there exists some index such that, for and are greater than
The normed space is called a Banach space and the canonical metric is called a complete metric if is a complete metric space, which by definition means for every Cauchy sequence in there exists some such that
where because this sequence's convergence to can equivalently be expressed as
The norm of a normed space is called a if is a Banach space.
L-semi-inner product
For any normed space there exists an L-semi-inner product on such that for all In general, there may be infinitely many L-semi-inner products that satisfy this condition and the proof of the existence of L-semi-inner products relies on the non-constructive Hahn–Banach theorem. L-semi-inner products are a generalization of inner products, which are what fundamentally distinguish Hilbert spaces from all other Banach spaces. This shows that all normed spaces (and hence all Banach spaces) can be considered as being generalizations of (pre-)Hilbert spaces.
Characterization in terms of series
The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space is a Banach space if and only if each absolutely convergent series in converges to a value that lies within see Theorem 1.3.9, p. 20 in . symbolically
Topology
The canonical metric of a normed space induces the usual metric topology on which is referred to as the canonical or norm induced topology. Every normed space is automatically assumed to carry this Hausdorff topology, unless indicated otherwise. With this topology, every Banach space is a Baire space, although there exist normed spaces that are Baire but not Banach. The norm is always a continuous function with respect to the topology that it induces.
The open and closed balls of radius centered at a point are, respectively, the sets
Any such ball is a convex and bounded subset of but a compact ball/neighborhood exists if and only if is finite-dimensional. In particular, no infinite–dimensional normed space can be locally compact or have the Heine–Borel property. If is a vector and is a scalar, then
Using shows that the norm-induced topology is translation invariant, which means that for any and the subset is open (respectively, closed) in if and only if its translation is open (respectively, closed). Consequently, the norm induced topology is completely determined by any neighbourhood basis at the origin. Some common neighborhood bases at the origin include
where can be any sequence of positive real numbers that converges to in (common choices are or ). So, for example, any open subset of can be written as a union
indexed by some subset where each may be chosen from the aforementioned sequence (The open balls can also be replaced with closed balls, although the indexing set and radii may then also need to be replaced). Additionally, can always be chosen to be countable if is a , which by definition means that contains some countable dense subset.
Homeomorphism classes of separable Banach spaces
All finite–dimensional normed spaces are separable Banach spaces and any two Banach spaces of the same finite dimension are linearly homeomorphic. Every separable infinite–dimensional Hilbert space is linearly isometrically isomorphic to the separable Hilbert sequence space with its usual norm
The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space of countably many copies of (this homeomorphism need not be a linear map). Thus all infinite–dimensional separable Fréchet spaces are homeomorphic to each other (or said differently, their topology is unique up to a homeomorphism). Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including In fact, is even homeomorphic to its own unit which stands in sharp contrast to finite–dimensional spaces (the Euclidean plane is not homeomorphic to the unit circle, for instance).
This pattern in homeomorphism classes extends to generalizations of metrizable (locally Euclidean) topological manifolds known as , which are metric spaces that are around every point, locally homeomorphic to some open subset of a given Banach space (metric Hilbert manifolds and metric Fréchet manifolds are defined similarly). For example, every open subset of a Banach space is canonically a metric Banach manifold modeled on since the inclusion map is an open local homeomorphism. Using Hilbert space microbundles, David Henderson showed in 1969 that every metric manifold modeled on a separable infinite–dimensional Banach (or Fréchet) space can be topologically embedded as an subset of and, consequently, also admits a unique smooth structure making it into a Hilbert manifold.
Compact and convex subsets
There is a compact subset of whose convex hull is closed and thus also compact.Let be the separable Hilbert space of square-summable sequences with the usual norm and let be the standard orthonormal basis (that is, each has zeros in every position except for a in the th-position). The closed set is compact (because it is sequentially compact) but its convex hull is a closed set because the point belongs to the closure of in but (since every point is a finite convex combination of elements of and so for all but finitely many coordinates, which is not true of ). However, like in all complete Hausdorff locally convex spaces, the convex hull of this compact subset is compact. The vector subspace is a pre-Hilbert space when endowed with the substructure that the Hilbert space induces on it, but is not complete and (since ). The closed convex hull of in (here, "closed" means with respect to and not to as before) is equal to which is not compact (because it is not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of a compact subset might to be compact (although it will be precompact/totally bounded). However, like in all Banach spaces, the convex hull of this (and every other) compact subset will be compact. In a normed space that is not complete then it is in general guaranteed that will be compact whenever is; an example can even be found in a (non-complete) pre-Hilbert vector subspace of
As a topological vector space
This norm-induced topology also makes into what is known as a topological vector space (TVS), which by definition is a vector space endowed with a topology making the operations of addition and scalar multiplication continuous. It is emphasized that the TVS is a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is associated with particular norm or metric (both of which are "forgotten"). This Hausdorff TVS is even locally convex because the set of all open balls centered at the origin forms a neighbourhood basis at the origin consisting of convex balanced open sets. This TVS is also , which by definition refers to any TVS whose topology is induced by some (possibly unknown) norm. Normable TVSs are characterized by being Hausdorff and having a bounded convex neighborhood of the origin. All Banach spaces are barrelled spaces, which means that every barrel is neighborhood of the origin (all closed balls centered at the origin are barrels, for example) and guarantees that the Banach–Steinhaus theorem holds.
Comparison of complete metrizable vector topologies
The open mapping theorem implies that when and are topologies on that make both and into complete metrizable TVSes (for example, Banach or Fréchet spaces), if one topology is finer or coarser than the other, then they must be equal (that is, if or then ). So, for example, if and are Banach spaces with topologies and and if one of these spaces has some open ball that is also an open subset of the other space (or, equivalently, if one of or is continuous), then their topologies are identical and the norms and are equivalent.
Completeness
Complete norms and equivalent norms
Two norms, and on a vector space are said to be equivalent if they induce the same topology; this happens if and only if there exist real numbers such that for all If and are two equivalent norms on a vector space then is a Banach space if and only if is a Banach space. See this footnote for an example of a continuous norm on a Banach space that is equivalent to that Banach space's given norm.Let denote the Banach space of continuous functions with the supremum norm and let denote the topology on induced by The vector space can be identified (via the inclusion map) as a proper dense vector subspace of the space which satisfies for all Let denote the restriction of to which makes this map a norm on (in general, the restriction of any norm to any vector subspace will necessarily again be a norm). The normed space is a Banach space since its completion is the proper superset Because holds on the map is continuous. Despite this, the norm is equivalent to the norm (because is complete but is not). All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.see Corollary 1.4.18, p. 32 in .
Complete norms vs complete metrics