Banach space

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 pair
consisting of a vector space over a scalar field together with a distinguished
norm Like all norms, this norm induces a translation invariant
distance function, called the canonical or induced metric, defined for all vectors by
This makes into a metric space
A sequence is called or or if for every real there exists some index such that
whenever 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 can be considered as being generalizations of 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 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 in if and only if its translation is open.
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 .
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 .
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 .
Thus all infinite–dimensional separable Fréchet spaces are homeomorphic to each other.
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.
This pattern in homeomorphism classes extends to generalizations of metrizable topological manifolds known as, which are metric spaces that are around every point, locally homeomorphic to some open subset of a given Banach space.
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 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.
However, like in all Banach spaces, the convex hull of this 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 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, 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. 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 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 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, if one topology is finer or coarser than the other, then they must be equal.
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, 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.
All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.

Complete norms vs complete metrics

A metric on a vector space is induced by a norm on if and only if is translation invariant and absolutely homogeneous, which means that for all scalars and all in which case the function defines a norm on and the canonical metric induced by is equal to
Suppose that is a normed space and that is the norm topology induced on Suppose that is metric on such that the topology that induces on is equal to If is translation invariant then is a Banach space if and only if is a complete metric space.
If is translation invariant, then it may be possible for to be a Banach space but for to be a complete metric space. In contrast, a theorem of Klee, which also applies to all metrizable topological vector spaces, implies that if there exists complete metric on that induces the norm topology on then is a Banach space.
A Fréchet space is a locally convex topological vector space whose topology is induced by some translation-invariant complete metric.
Every Banach space is a Fréchet space but not conversely; indeed, there even exist Fréchet spaces on which no norm is a continuous function.
However, the topology of every Fréchet space is induced by some countable family of real-valued maps called seminorms, which are generalizations of norms.
It is even possible for a Fréchet space to have a topology that is induced by a countable family of
but to not be a Banach/normable space because its topology can not be defined by any norm.
An example of such a space is the Fréchet space whose definition can be found in the article on spaces of test functions and distributions.

Complete norms vs complete topological vector spaces

There is another notion of completeness besides metric completeness and that is the notion of a complete topological vector space or TVS-completeness, which uses the theory of uniform spaces.
Specifically, the notion of TVS-completeness uses a unique translation-invariant uniformity, called the canonical uniformity, that depends on vector subtraction and the topology that the vector space is endowed with, and so in particular, this notion of TVS completeness is independent of whatever norm induced the topology .
Every Banach space is a complete TVS. Moreover, a normed space is a Banach space if and only if it is complete as a topological vector space.
If is a metrizable topological vector space, then is a complete TVS if and only if it is a complete TVS, meaning that it is enough to check that every Cauchy in converges in to some point of .
If is a topological vector space whose topology is induced by norm, then is a complete topological vector space if and only if may be assigned a norm that induces on the topology and also makes into a Banach space.
A Hausdorff locally convex topological vector space is normable if and only if its strong dual space is normable, in which case is a Banach space.
If is a metrizable locally convex TVS, then is normable if and only if is a Fréchet–Urysohn space.
This shows that in the category of locally convex TVSs, Banach spaces are exactly those complete spaces that are both metrizable and have metrizable strong dual spaces.