# Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) 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.[1] Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space."[2] 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.

A linear functional on a normed space is a bounded linear functional if and only if it is a continuous function. More generally, a linear operator between two normed spaces is bounded if and only if it is continuous. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is often used when dealing with Banach spaces.

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

For Y a Banach space, the space B(X, Y) is a Banach space with respect to this norm.

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

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

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.[21]

A subset S in a Banach space X is total if the linear span of S is dense in X. The subset S is total in X if and only if the only continuous linear functional that vanishes on S is the 0 functional: this equivalence follows from the Hahn–Banach theorem.

If X is the direct sum of two closed linear subspaces M and N, then the dual X ′ of X is isomorphic to the direct sum of the duals of M and N.[22] If M is a closed linear subspace in X, one can associate the orthogonal of M in the dual,

When X ′ is separable, the above criterion for totality can be used for proving the existence of a countable total subset in X.

The Banach–Alaoglu theorem can be proved using Tychonoff's theorem about infinite products of compact Hausdorff spaces. When X is separable, the unit ball B ′ of the dual is a metrizable compact in the weak* topology.[27]

When K is a compact Hausdorff topological space, the dual M(K) of C(K) is the space of Radon measures in the sense of Bourbaki.[28] The subset P(K) of M(K) consisting of non-negative measures of mass 1 (probability measures) is a convex w*-closed subset of the unit ball of M(K). The extreme points of P(K) are the Dirac measures on K. The set of Dirac measures on K, equipped with the w*-topology, is homeomorphic to K.

The result has been extended by Amir[31] and Cambern[32] to the case when the multiplicative Banach–Mazur distance between C(K) and C(L) is < 2. The theorem is no longer true when the distance is = 2.[33]

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 and the latter with the w*-topology. In this identification, the maximal ideal space can be viewed as a w*-compact subset of the unit ball in the dual A ′.

Not every unital commutative Banach algebra is of the form C(K) for some compact Hausdorff space K. However, this statement holds if one places C(K) in the smaller category of commutative C*-algebras. Gelfand's representation theorem for commutative C*-algebras states that every commutative unital C*-algebra A is isometrically isomorphic to a C(K) space.[34] The Hausdorff compact space K here is again the maximal ideal space, also called the spectrum of A in the C*-algebra context.

Here are the main general results about Banach spaces that go back to the time of Banach's book (Banach (1932)) 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.

The Banach–Steinhaus theorem is not limited to Banach spaces. It can be extended for example to the case where X is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood U of 0 in X such that all T in F are uniformly bounded on U,

Corollary. Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism.

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

This is a consequence of the Hahn–Banach theorem. Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space X onto the Banach space Y, then Y is reflexive.

Indeed, if the dual Y ′ of a Banach space Y is separable, then Y is separable. If X is reflexive and separable, then the dual of X ′ is separable, so X ′ is separable.

When X is reflexive, it follows that all closed and bounded convex subsets of X are weakly compact. In a Hilbert space H, the weak compactness of the unit ball is very often used in the following way: every bounded sequence in H 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 continuous function on the unit ball B of a reflexive space attains its minimum at some point in B.

James' Theorem. For a Banach space the following two properties are equivalent:

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

On every non-reflexive Banach space X, there exist continuous linear functionals that are not norm-attaining. However, the BishopPhelps theorem[37] states that norm-attaining functionals are norm dense in the dual X ′ of X.

When the Banach space X is separable, the unit ball of the dual X ′, equipped with the weak*-topology, is a metrizable compact space K,[27] and every element x ′′ in the bidual X ′′ defines a bounded function on K:

When X is separable, the unit ball of the dual is weak*-compact by Banach–Alaoglu and metrizable for the weak* topology,[27] 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.

The weak topology of a Banach space X is metrizable if and only if X is finite-dimensional.[46] If the dual X ′ is separable, the weak topology of the unit ball of X 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 X is reflexive if and only if each bounded sequence in X has a weakly convergent subsequence.[48]

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

Since every vector x in a Banach space X with a basis is the limit of Pn(x), with Pn of finite rank and uniformly bounded, the space X 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.[52]

Robert C. James characterized reflexivity in Banach spaces with a basis: the space X with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete.[53] In this case, the biorthogonal functionals form a basis of the dual of X.

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.[54]

A necessary and sufficient condition for the norm of a Banach space X to be associated to an inner product is the parallelogram identity:

To see that the parallelogram law is sufficient, one observes in the real case that < x, y > is symmetric, and in the complex case, that it satisfies the Hermitian symmetry property and < ix, y > = i < x, y >. The parallelogram law implies that < x, y > is additive in x. It follows that it is linear over the rationals, thus linear by continuity.

Several characterizations of spaces isomorphic (rather than isometric) to Hilbert spaces are available. The parallelogram law can be extended to more than two vectors, and weakened by the introduction of a two-sided inequality with a constant c ≥ 1: Kwapień proved that if

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.[62] The proof rests upon Dvoretzky's theorem about Euclidean sections of high-dimensional centrally symmetric convex bodies. In other words, Dvoretzky's theorem states that for every integer n, any finite-dimensional normed space, with dimension sufficiently large compared to n, contains subspaces nearly isometric to the n-dimensional Euclidean space.

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[67] that any two infinite-dimensional separable Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved[68] that any two Banach spaces are homeomorphic if and only if they have the same density character, the minimum cardinality of a dense subset.

The situation is different for countably infinite compact Hausdorff spaces. Every countably infinite compact K is homeomorphic to some closed interval of ordinal numbers

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.