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) together with a distinguished norm Like all norms, this norm induces a translation invariant distance function, called the canonical or (norm) induced metric by into a metric space is called or or is called a Banach space is called a complete metric is a complete metric space, which by definition means for every Cauchy sequence

is called a is a Banach space.

L-semi-inner product

there exists an L-semi-inner product 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 is a Banach space if and only if each absolutely convergent

Topology

induces the usual metric topology 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. is always a continuous function with respect to the topology that it induces.

Any such ball is a convex and bounded subset but a compact ball/neighborhood is finite-dimensional. In particular, no infinite–dimensional normed space can be locally compact or have the Heine–Borel property shows that the norm-induced topology is translation invariant is open (respectively, closed is open (respectively, closed). Consequently, the norm induced topology is completely determined by any neighbourhood basis can always be chosen to be countable is a 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 \ell^2(\N)

The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space (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). is even homeomorphic to its own unit sphere 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). 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 open subset and, consequently, also admits a unique smooth structure Hilbert manifold.

Compact and convex subsets

whose convex hull is closed and thus also compact. However, like in all Banach spaces, the closed 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 is; an example can even be found in a (non-complete) pre-Hilbert

As a topological vector space

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 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 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 into complete metrizable TVSeses (for example, Banach or Fréchet spaces), if one topology is finer or coarser). are equivalent.

Completeness

Complete norms and equivalent norms

are said to be equivalent if they induce the same topology; 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

is translation invariant and absolutely homogeneous

is metric is translation invariant is a complete metric space. is to be a complete metric space (see this footnote for an example). In contrast, a theorem of Klee, which also applies to all metrizable topological vector spaces, implies that if there exists 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 (such as the space of real sequences with the product topology). However, the topology of every Fréchet space is induced by some countable family of real-valued (necessarily continuous) maps called seminorms, which are generalizations of normss. It is even possible for a Fréchet space to have a topology that is induced by a countable family of (such norms would necessarily be continuous) 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 (TVS) 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 (and even applies to TVSs that are is a metrizable topological vector space is a complete TVS if and only if it is a complete TVS, meaning that it is enough to check that every Cauchy (that is, there is no need to consider the more general notion of arbitrary Cauchy nets).

is a topological vector space whose topology is induced by (possibly unknown) norm (such spaces are called may be assigned a norm into a Banach space. A Hausdorff locally convex topological vector space is normable if and only if its strong dual space is normable, denotes the strong dual space whose topology is a generalization of the dual norm-induced topology on the continuous dual space; see this footnote is a metrizable 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.

Completions

Every normed space can be isometrically embedded onto a dense vector subspace of a Banach space, where this Banach space is called a completion of the normed space. This Hausdorff completion is unique up to isometric isomorphism.

is an isometric mapping is the Hausdorff completion

General theory

Linear operators, isomorphisms

are normed spaces over the same ground field the set of all continuous \mathbb{K}-linear maps to another normed space is continuous if and only if it is bounded on the closed unit ball can be given the operator norm

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 reappears as a natural bifunctor.

forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.

are normed spaces, they are isomorphic normed spaces is complete (or reflexive, separable are isometrically isomorphic is an isometry The Banach–Mazur distance differ.

Continuous and bounded linear functions and seminorms

Every continuous linear operator is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is, a linear operator between two normed spaces is bounded if and only if it is a continuous function) is a normed space, a linear functional on a normed space is a bounded linear functional if and only if it is a continuous linear functional. This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used when dealing primarily with Banach spaces.

is a subadditive function (such as a norm, a sublinear function, or real linear functional), then is continuous at the origin is uniformly continuous is continuous if and only if its absolute value And very importantly for applying the Hahn–Banach theorem is continuous if and only if this is true of its real part and \operatorname{Re} f completely determines is continuous if and only if the seminorm is encountered in many versions of the Hahn–Banach theorem.

Basic notions

of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used, which correspond (respectively) to the coproduct and product in the category of Banach spaces and short maps (discussed above).) is complete if and only if the two factors are complete.

is a closed linear subspace

is complete. in which case the quotient is the null space.

is said to be a complemented subspace is the range of a surjective bounded linear projection

as is a linear bijection whose inverse need not be bounded.

Classical spaces

Basic examples of Banach spaces include: the Lp spaces and their special cases, the sequence spaces that consist of scalar sequences indexed by natural numbers of absolutely summable

According to the Banach–Mazur theorem

Any Hilbert space is the inner product

is a Hilbert space.

The Hardy spaces, the Sobolev spaces spaces and have additional structure. They are important in different branches of analysis, Harmonic analysis and Partial differential equations among others.

Banach algebras

A Banach algebra together with a structure of \mathbb{K}

Examples

• with the pointwise product, is a Banach algebra.

• The disk algebra consists of functions holomorphic in the open unit disk and continuous on its closure

• The Wiener algebra with absolutely convergent Fourier series where the product is the convolution of sequences.

• with the composition of maps as product, is a Banach algebra.

• A C*-algebra with an antilinear involution is a fundamental example of C*-algebra. The Gelfand–Naimark theorem its complex conjugate

Dual space

the underlying field (either the realss or the complex numbers), the continuous dual space or continuous linear functionals in this article. is a Banach space (using the absolute value The Dixmier–Ng theorem characterizes the dual spaces of Banach spaces.

The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem.

In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional.

must have norm one, and is called a norming functional

The Hahn–Banach separation theorem states that two disjoint non-empty convex sets in a real Banach space, one of them open, can be separated by a closed affine hyperplane. The open convex set lies strictly on one side of the hyperplane, the second convex set lies on the other side but may touch the hyperplane.

is total if the linear span is dense functional: this equivalence follows from the Hahn–Banach theorem.

one can associate the

The dual of a separable Banach space need not be separable, but:

Weak topologies

The weak topology is the coarsest topology are continuous. The norm topology is therefore finer than the weak topology. It follows from the Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a norm-closed convex subset of a Banach space is also weakly closed. is also weakly continuous

is called the algebraic dual space which is finer than the weak topology, and much less used in functional analysis.

called the weak* topology are continuous. Its importance comes from the Banach–Alaoglu theorem.

The Banach–Alaoglu theorem can be proved using Tychonoff's theorem of the dual is a metrizable compact in the weak* topology.

Examples of dual spaces

. The dual of Lebesgue space

The Riesz representation theorem is an antilinear When the scalars are real, this map is an isometric isomorphism.

is the space of Radon measures in the sense of Bourbaki. consisting of non-negative measures of mass 1 (probability measures The extreme points are the Dirac measures equipped with the w*-topology, is homeomorphic

The result has been extended by Amir and Cambern to the case when the multiplicative Banach–Mazur distance

In the commutative Banach algebra the maximal ideals

More generally, by the Gelfand–Mazur theorem, the maximal ideals of a unital commutative Banach algebra can be identified with its characters—not merely as sets but as topological spaces: the former with the hull-kernel topology

in the smaller category of commutative C*-algebras. Gelfand's representation theorem for commutative C*-algebras states that every commutative unital C space. here is again the maximal ideal space, also called the spectrum in the C*-algebra context.

Bidual

is called the or

As a consequence of the existence of a norming functional is isometric, thus injective.

It is indeed isometric, but not onto.

is surjective is called reflexive (see below is complete, therefore, every reflexive normed space is a Banach space.

The Goldstine theorem in the bidual, there exists a net

cannot be weak*-limit of is weakly sequentially complete.

Banach's theorems

Here are the main general results about Banach spaces that go back to the time of Banach's book () and are related to the Baire category theorem. According to this theorem, a complete metric space (such as a Banach space, a Fréchet space or an F-space) cannot be equal to a union of countably many closed subsets with empty interiors. Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional.

is a Fréchet space

This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorization of bounded linear maps.

Reflexivity

is called reflexive is surjective. Reflexive normed spaces are Banach spaces.

is reflexive.

is separable.

More generally, uniformly convex spaces are reflexive, by the Milman–Pettis theorem is infinite-dimensional, and even nonseparable. However, Robert C. James has constructed an example of a non-reflexive space, usually called "the James space is isometrically isomorphic to its bidual.

is reflexive, it follows that all closed and bounded convex subsets has weakly convergent subsequences.

Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain optimization problems. For example, every convex

The following theorem of Robert C. James provides a converse statement.

The theorem can be extended to give a characterization of weakly compact convex sets.

there exist continuous linear functionals that are not norm-attaining. However, the Bishop–Phelps theorem

Weak convergences of sequences

is weakly convergent is a weakly Cauchy sequence is weakly* convergent Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the Banach–Steinhaus theorem.

is weakly sequentially complete It follows from the preceding discussion that reflexive spaces are weakly sequentially complete.

vector. The unit vector basis is another example of a weakly null sequence

are norm convergent. satisfies Schur's property.

Results involving the 𝓁1 basis

basis are the opposite cases of the dichotomy established in the following deep result of Haskell P. Rosenthal.

A complement to this result is due to Odell and Rosenthal(1975).

is weak*-limit of a

is the pointwise limit it is therefore a first Baire class function

Sequences, weak and weak* compactness

is separable, the unit ball of the dual is weak*-compact by the Banach–Alaoglu theorem and metrizable for the weak* topology, hence every bounded sequence in the dual has weakly* convergent subsequences. This applies to separable reflexive spaces, but more is true in this case, as stated below.

is finite-dimensional. 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.

has a weakly convergent subsequence.

Type and cotype

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.

Schauder bases

A Schauder basis there exist

Banach spaces with a Schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients (say) is dense.

in the above expansion. They are called biorthogonal functionals

Most classical separable spaces have explicit bases. The Haar system The trigonometric system The Schauder system has a basis admits a basis constructed from the Franklin system.

satisfies the bounded approximation property. The first example by Enflo of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis.

with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete.

Tensor product

-vector spaces. The tensor product which has the following universal property:

and called a simple tensor is a finite sum of such simple tensors.

There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm introduced by A. Grothendieck in 1955.

In general, the tensor product of complete spaces is not complete again. When working with Banach spaces, it is customary to say that the projective tensor product is the equipped with the projective tensor norm, and similarly for the injective tensor product Grothendieck proved in particular that

Tensor products and the approximation property

has the approximation property, this closure coincides with the space of compact operators

obtained by extending the identity map of the algebraic tensor product. Grothendieck related the approximation problem has the approximation property.

are infinite-dimensional Banach spaces. This was disproved by Gilles Pisier in 1983. are equal. Furthermore, just as Enflo's does not have the approximation property.

Some classification results

Characterizations of Hilbert space among Banach spaces

to be associated to an inner product is the parallelogram identity:

It follows, for example, that the Lebesgue space If this identity is satisfied, the associated inner product is given by the polarization identity

For complex scalars, defining the inner product antilinear

is symmetric, and in the complex case, that it satisfies the Hermitian symmetry It follows that it is linear over the rationals, thus linear by continuity.

is isomorphic to a Hilbert space. In the same article, Kwapień proved that the validity of a Banach-valued Parseval's theorem for the Fourier transform characterizes Banach spaces isomorphic to Hilbert spaces.

Lindenstrauss and Tzafriri proved that a Banach space in which every closed linear subspace is complemented (that is, is the range of a bounded linear projection) is isomorphic to a Hilbert space. The proof rests upon Dvoretzky's theorem-dimensional Euclidean space.

The next result gives the solution of the so-called is said to be homogeneous is homogeneous, and Banach asked for the converse.

An infinite-dimensional Banach space is hereditarily indecomposable when no subspace of it can be isomorphic to the direct sum of two infinite-dimensional Banach spaces. The Gowers dichotomy theorem with unconditional basis is not isomorphic to its closed hyperplanes. is homogeneous, it must therefore have an unconditional basis. It follows then from the partial solution obtained by Komorowski and Tomczak–Jaegermann, for spaces with an unconditional basis,

Metric classification

is an isometry), then the Mazur–Ulam theorem must be linear. This result implies that the metric in Banach spaces, and more generally in normed spaces, completely captures their linear structure.

Topological classification

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.

Spaces of continuous functions

are homeomorphic see above the results by Amir and Cambern. Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:

The situation is different for countably infinite is homeomorphic to some closed interval of ordinal numbers equipped with the order topology is a countably infinite ordinal. is then isometric to the spaces and are isomorphic if and only if . are mutually non-isomorphic.

Examples

Derivatives

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.

Generalizations

or the space of all distributions 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.

See also

• Banach-Saks property

Notes

References

Bibliography

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External links