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.

DefinitionEdit

A Banach space is a complete normed space   A normed space is a pair[note 1]   consisting of a vector space   over a scalar field K (where K is   or  ) together with a distinguished[note 2] norm   Like all norms, this norm induces a translation invariant[note 3] distance function, called the canonical or (norm) induced metric, defined by[note 4]

 
for all vectors   This makes   a metric space   A sequence   is called  -Cauchy or Cauchy in  [note 5] or  -Cauchy if and only if for every real   there exists some index   such that
 
whenever   and   are greater than   The canonical metric   is called a complete metric if the pair   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 can equivalently be expressed as:
 

By definition, the normed space   is a Banach space if and only if   is a complete metric space, or said differently, if and only if the canonical metric   is a complete metric. The norm   of a normed space   is called a complete norm if and only if   is a Banach space.

L-semi-inner product

For any normed space   there exists an L-semi-inner product ("L" is for Günter Lumer)   on   such that   for all  ; in general, there may be infinitely many L-semi-inner products that satisfy this condition. 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.

TopologyEdit

The canonical metric   of a normed space   induces the usual metric topology   on   where this topology, which is referred to as the canonical or norm induced topology, makes   into a Hausdorff metrizable topological space. Every normed space is automatically assumed to carry this topology, unless indicated otherwise. With this topology, every Banach space is a Baire space, although there are normed spaces that are Baire but not Banach.[3]

This norm-induced topology is translation invariant, which means that for any   and   the subset   is open (resp. closed) in   if and only if this is true of   Consequently, the norm induced topology is completely determined by any neighbourhood basis at the origin. Some common neighborhood bases at the origin include:

 
where   is any sequence in of positive real numbers that converges to   in   (such as   for instance) and where
 
are respectively, the open ball and closed ball of radius   centered at the origin. So for example, every open subset   of   can be written as a union   indexed by some subset   where every   is an integer; the closed ball   centered at the origin can also be used instead of the open ball (although the subset   and integers   might need to be changed).

This norm-induced topology also makes   into what is known as a topological vector space (TVS),[note 6] 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 only a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is not associated with any particular norm or metric (both of which are "forgotten").

CompletenessEdit

Complete norms and equivalent norms

Two norms on a vector space are called equivalent if and only if they induce the same topology.[4] 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 not equivalent to that Banach space's given norm.[note 7][4] All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.[5]

Complete norms vs complete metrics

A metric   on a vector space   is induced by a norm on   if and only if   is translation invariant[note 3] 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 any metric on   such that the topology that   induces on   is equal to   If   is translation invariant[note 3] then   is a Banach space if and only if   is a complete metric space.[6] If   is not translation invariant, then it may be possible for   to be a Banach space but   to not be a complete metric space[7] (see this footnote[note 8] for an example). In contrast, a theorem of Klee,[8][9][note 9] which also applies to all metrizable topological vector spaces, implies that if there exists any[note 10] complete metric   on   that induces the norm topology on   then   is a Banach space.

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 only 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   (and even applies to TVSs that are not even metrizable). Every Banach space is a complete TVS. Moreover, a normed space is a Banach space (i.e. its norm-induced metric is complete) if and only if it is complete as a topological vector space. If   is a metrizable topological vector space (where note that every norm induced topology is metrizable), then   is a complete TVS if and only if it is a sequentially complete TVS, meaning that it is enough to check that every Cauchy sequence in   converges in   to some point of   (that is, there is no need to consider the more general notion of arbitrary Cauchy nets).

If   is a topological vector space whose topology is induced by some (possibly unknown) 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,[10] in which case   is a Banach space (  denotes the strong dual space of   whose topology is a generalization of the dual norm-induced topology on the continuous dual space  ; see this footnote[note 11] for more details). If   is a metrizable locally convex TVS, then   is normable if and only if   is a Fréchet–Urysohn space.[11] 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.

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 in  [12]

 

CompletionsEdit

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.

More precisely, for every normed space   there exist a Banach space   and a mapping   such that   is an isometric mapping and   is dense in   If   is another Banach space such that there is an isometric isomorphism from   onto a dense subset of   then   is isometrically isomorphic to   This Banach space   is the completion of the normed space   The underlying metric space for   is the same as the metric completion of   with the vector space operations extended from   to   The completion of   is often denoted by  

General theoryEdit

Linear operators, isomorphismsEdit

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

 

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

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

If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection   such that T and its inverse   are continuous. If one of the two spaces X or Y is complete (or reflexive, separable, etc.) then so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry, that is,   for every x in X. The Banach–Mazur distance   between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and Y differ.

Continuous and bounded linear functions and seminormsEdit

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. So in particular, because the scalar field (which is   or  ) 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.

If   is a subadditive function (such as a norm, a sublinear function, or real linear functional), then[13]   is continuous at the origin if and only if   is uniformly continuous on all of  ; and if in addition   then   is continuous if and only if its absolute value   is continuous, which happens if and only if   is an open subset of  [13][note 12] By applying this to   it follows that the norm   is always a continuous map. And very importantly for applying the Hahn-Banach theorem, a linear functional   is continuous if and only if this is true of its real part   and moreover,   and the real part   completely determines   which is why the Hahn-Banach theorem is often stated only for real linear functionals. Also, a linear functional   on   is continuous if and only if the seminorm   is continuous, which happens if and only if there exists a continuous seminorm   such that  ; this last statement involving the linear functional   and seminorm   is encountered in many versions of the Hahn-Banach theorem.

Basic notionsEdit

The Cartesian product   of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used,[14] such as

 
and give rise to isomorphic normed spaces. In this sense, the product   (or the direct sum  ) is complete if and only if the two factors are complete.

If M is a closed linear subspace of a normed space   there is a natural norm on the quotient space  

 

The quotient   is a Banach space when   is complete.[15] The quotient map from   onto   sending   to its class   is linear, onto and has norm 1, except when   in which case the quotient is the null space.

The closed linear subspace   of   is said to be a complemented subspace of   if   is the range of a surjective bounded linear projection   In this case, the space   is isomorphic to the direct sum of M and   the kernel of the projection  

Suppose that   and   are Banach spaces and that   There exists a canonical factorization of   as[15]

 
where the first map   is the quotient map, and the second map   sends every class   in the quotient to the image   in   This is well defined because all elements in the same class have the same image. The mapping   is a linear bijection from   onto the range   whose inverse need not be bounded.

Classical spacesEdit

Basic examples[16] of Banach spaces include: the Lp spaces   and their special cases, the sequence spaces   that consist of scalar sequences indexed by natural numbers  ; among them, the space   of absolutely summable sequences and the space   of square summable sequences; the space   of sequences tending to zero and the space   of bounded sequences; the space   of continuous scalar functions on a compact Hausdorff space   equipped with the max norm,

 

According to the Banach–Mazur theorem, every Banach space is isometrically isomorphic to a subspace of some  [17] For every separable Banach space X, there is a closed subspace   of   such that  [18]

Any Hilbert space serves as an example of a Banach space. A Hilbert space   on   is complete for a norm of the form

 
where
 
is the inner product, linear in its first argument that satisfies the following:
 

For example, the space   is a Hilbert space.

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

Banach algebrasEdit

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  

ExamplesEdit

  • The Banach space   with the pointwise product, is a Banach algebra.
  • The disk algebra A(D) consists of functions holomorphic in the open unit disk   and continuous on its closure: D. Equipped with the max norm on D, the disk algebra A(D) is a closed subalgebra of C(D).
  • The Wiener algebra A(T) is the algebra of functions on the unit circle T with absolutely convergent Fourier series. Via the map associating a function on T to the sequence of its Fourier coefficients, this algebra is isomorphic to the Banach algebra   where the product is the convolution of sequences.
  • For every Banach space X, the space B(X) of bounded linear operators on X, with the composition of maps as product, is a Banach algebra.
  • A C*-algebra is a complex Banach algebra A with an antilinear involution   such that   The space B(H) of bounded linear operators on a Hilbert space H is a fundamental example of C*-algebra. The Gelfand–Naimark theorem states that every C*-algebra is isometrically isomorphic to a C*-subalgebra of some B(H). The space   of complex continuous functions on a compact Hausdorff space   is an example of commutative C*-algebra, where the involution associates to every function   its complex conjugate  

Dual spaceEdit

If X is a normed space and   the underlying field (either the real or the complex numbers), the continuous dual space is the space of continuous linear maps from X into   or continuous linear functionals. The notation for the continuous dual is   in this article.[19] Since   is a Banach space (using the absolute value as norm), the dual X ′ is a Banach space, for every normed space X.

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

Hahn–Banach theorem. Let X be a vector space over the field   Let further
  •   be a linear subspace,
  •   be a sublinear function and
  •   be a linear functional so that   for all y in Y.
Then, there exists a linear functional  } so that
 

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.[20] An important special case is the following: for every vector x in a normed space X, there exists a continuous linear functional   on X such that

 

When x is not equal to the 0 vector, the functional   must have norm one, and is called a norming functional for x.

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,

 

The orthogonal   is a closed linear subspace of the dual. The dual of M is isometrically isomorphic to   The dual of   is isometrically isomorphic to  [23]

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

Theorem.[24] Let X be a normed space. If X ′ is separable, then X is separable.

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

Weak topologiesEdit

The weak topology on a Banach space X is the coarsest topology on X for which all elements   in the continuous dual space   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.[25] A norm-continuous linear map between two Banach spaces X and Y is also weakly continuous, i.e., continuous from the weak topology of X to that of Y.[26]

If X is infinite-dimensional, there exist linear maps which are not continuous. The space   of all linear maps from X to the underlying field   (this space   is called the algebraic dual space, to distinguish it from   also induces a topology on X which is finer than the weak topology, and much less used in functional analysis.

On a dual space  , there is a topology weaker than the weak topology of X ′, called weak* topology. It is the coarsest topology on   for which all evaluation maps   where   ranges over   are continuous. Its importance comes from the Banach–Alaoglu theorem.

Banach–Alaoglu Theorem. Let X be a normed vector space. Then the closed unit ball   of the dual space is compact in the weak* topology.

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]

Examples of dual spacesEdit

The dual of   is isometrically isomorphic to  : for every bounded linear functional   on   there is a unique element   such that

 

The dual of   is isometrically isomorphic to  }. The dual of Lebesgue space   is isometrically isomorphic to   when   and  

For every vector   in a Hilbert space   the mapping

 

defines a continuous linear functional   on  The Riesz representation theorem states that every continuous linear functional on H is of the form   for a uniquely defined vector   in   The mapping   is an antilinear isometric bijection from H onto its dual H ′. When the scalars are real, this map is an isometric isomorphism.

When   is a compact Hausdorff topological space, the dual   of   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.

Banach–Stone Theorem. If K and L are compact Hausdorff spaces and if   and   are isometrically isomorphic, then the topological spaces K and L are homeomorphic.[29][30]

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

In the commutative Banach algebra   the maximal ideals are precisely kernels of Dirac measures on K,

 

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 ′.

Theorem. If K is a compact Hausdorff space, then the maximal ideal space Ξ of the Banach algebra   is homeomorphic to K.[29]

Not every unital commutative Banach algebra is of the form   for some compact Hausdorff space K. However, this statement holds if one places   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   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.

BidualEdit

If X is a normed space, the (continuous) dual   of the dual   is called bidual, or second dual of   For every normed space X, there is a natural map,

 

This defines   as a continuous linear functional on   that is, an element of   The map   is a linear map from X to   As a consequence of the existence of a norming functional   for every   this map   is isometric, thus injective.

For example, the dual of   is identified with   and the dual of   is identified with   the space of bounded scalar sequences. Under these identifications,   is the inclusion map from   to   It is indeed isometric, but not onto.

If   is surjective, then the normed space X is called reflexive (see below). Being the dual of a normed space, the bidual   is complete, therefore, every reflexive normed space is a Banach space.

Using the isometric embedding   it is customary to consider a normed space X as a subset of its bidual. When X is a Banach space, it is viewed as a closed linear subspace of   If X is not reflexive, the unit ball of X is a proper subset of the unit ball of   The Goldstine theorem states that the unit ball of a normed space is weakly*-dense in the unit ball of the bidual. In other words, for every   in the bidual, there exists a net   in X so that

 

The net may be replaced by a weakly*-convergent sequence when the dual   is separable. On the other hand, elements of the bidual of   that are not in   cannot be weak*-limit of sequences in   since   is weakly sequentially complete.

Banach's theoremsEdit

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.

Banach–Steinhaus Theorem. Let X be a Banach space and Y be a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y. The uniform boundedness principle states that if for all x in X we have   then  

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,

 
The Open Mapping Theorem. Let X and Y be Banach spaces and   be a surjective continuous linear operator, then T is an open map.
Corollary. Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism.
The First Isomorphism Theorem for Banach spaces. Suppose that X and Y are Banach spaces and that   Suppose further that the range of T is closed in Y. Then   is isomorphic to  

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

Corollary. If a Banach space X is the internal direct sum of closed subspaces   then X is isomorphic to  

This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from   onto X sending   to the sum  

The Closed Graph Theorem. Let   be a linear mapping between Banach spaces. The graph of T is closed in   if and only if T is continuous.

ReflexivityEdit

The normed space X is called reflexive when the natural map

 
is surjective. Reflexive normed spaces are Banach spaces.
Theorem. If X is a reflexive Banach space, every closed subspace of X and every quotient space of X are reflexive.

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.

Theorem. If X is a Banach space, then X is reflexive if and only if X ′ is reflexive.
Corollary. Let X be a reflexive Banach space. Then X is separable if and only if X ′ is separable.

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.

Theorem. Suppose that   are normed spaces and that   Then X is reflexive if and only if each   is reflexive.

Hilbert spaces are reflexive. The Lp spaces are reflexive when   More generally, uniformly convex spaces are reflexive, by the Milman–Pettis theorem. The spaces   are not reflexive. In these examples of non-reflexive spaces X, the bidual X ′′ is "much larger" than X. Namely, under the natural isometric embedding of X into X ′′ given by the Hahn–Banach theorem, the quotient X ′′ / X is infinite-dimensional, and even nonseparable. However, Robert C. James has constructed an example[35] of a non-reflexive space, usually called "the James space" and denoted by J,[36] such that the quotient J ′′ / J is one-dimensional. Furthermore, this space J is isometrically isomorphic to its bidual.

Theorem. A Banach space X is reflexive if and only if its unit ball is compact in the weak topology.

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.

As a special case of the preceding result, when X is a reflexive space over   every continuous linear functional   in X ′ attains its maximum   on the unit ball of X. The following theorem of Robert C. James provides a converse statement.

James' Theorem. For a Banach space the following two properties are equivalent:
  • X is reflexive.
  • for all   in   there exists   with   so that  

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.

Weak convergences of sequencesEdit

A sequence   in a Banach space X is weakly convergent to a vector   if   converges to   for every continuous linear functional   in the dual X ′. The sequence   is a weakly Cauchy sequence if   converges to a scalar limit  , for every   in X ′. A sequence   in the dual X ′ is weakly* convergent to a functional   if   converges to   for every x in X. Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the Banach–Steinhaus theorem.

When the sequence   in X is a weakly Cauchy sequence, the limit L above defines a bounded linear functional on the dual X ′, that is, an element L of the bidual of X, and L is the limit of   in the weak*-topology of the bidual. The Banach space X is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent in X. It follows from the preceding discussion that reflexive spaces are weakly sequentially complete.

Theorem. [38] For every measure   the space   is weakly sequentially complete.

An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the 0 vector. The unit vector basis of   for   or of   is another example of a weakly null sequence, i.e., a sequence that converges weakly to 0. For every weakly null sequence in a Banach space, there exists a sequence of convex combinations of vectors from the given sequence that is norm-converging to 0.[39]

The unit vector basis of   is not weakly Cauchy. Weakly Cauchy sequences in   are weakly convergent, since L1-spaces are weakly sequentially complete. Actually, weakly convergent sequences in   are norm convergent.[40] This means that   satisfies Schur's property.

Results involving the   basisEdit

Weakly Cauchy sequences and the   basis are the opposite cases of the dichotomy established in the following deep result of H. P. Rosenthal.[41]

Theorem.[42] Let   be a bounded sequence in a Banach space. Either   has a weakly Cauchy subsequence, or it admits a subsequence equivalent to the standard unit vector basis of  

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

Theorem.[43] Let X be a separable Banach space. The following are equivalent:
  • The space X does not contain a closed subspace isomorphic to  
  • Every element of the bidual X ′′ is the weak*-limit of a sequence   in X.

By the Goldstine theorem, every element of the unit ball B ′′ of X ′′ is weak*-limit of a net in the unit ball of X. When X does not contain   every element of B ′′ is weak*-limit of a sequence in the unit ball of X.[44]

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:

 

This function is continuous for the compact topology of K if and only if x ′′ is actually in X, considered as subset of X ′′. Assume in addition for the rest of the paragraph that X does not contain   By the preceding result of Odell and Rosenthal, the function x ′′ is the pointwise limit on K of a sequence   of continuous functions on K, it is therefore a first Baire class function on K. The unit ball of the bidual is a pointwise compact subset of the first Baire class on K.[45]

Sequences, weak and weak* compactnessEdit

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.

Eberlein–Šmulian theorem.[47] A set A in a Banach space is relatively weakly compact if and only if every sequence {an} in A has a weakly convergent subsequence.

A Banach space X is reflexive if and only if each bounded sequence in X has a weakly convergent subsequence.[48]

A weakly compact subset A in   is norm-compact. Indeed, every sequence in A has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of  

Schauder basesEdit

A Schauder basis in a Banach space X is a sequence   of vectors in X with the property that for every vector x in X, there exist uniquely defined scalars   depending on x, such that

 

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

It follows from the Banach–Steinhaus theorem that the linear mappings {Pn} are uniformly bounded by some constant C. Let {e
n
}
denote the coordinate functionals which assign to every x in X the coordinate   of x in the above expansion. They are called biorthogonal functionals. When the basis vectors have norm 1, the coordinate functionals {e
n
}
have norm   in the dual of X.

Most classical separable spaces have explicit bases. The Haar system   is a basis for   The trigonometric system is a basis in Lp(T) when   The Schauder system is a basis in the space C([0, 1]).[49] The question of whether the disk algebra A(D) has a basis[50] remained open for more than forty years, until Bočkarev showed in 1974 that A(D) admits a basis constructed from the Franklin system.[51]

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.

Tensor productEdit

Let   and   be two  -vector spaces. The tensor product   of   and   is a  -vector space   with a bilinear mapping   which has the following universal property:

If   is any bilinear mapping into a  -vector space   then there exists a unique linear mapping   such that  

The image under   of a couple   in   is denoted by   and called a simple tensor. Every element   in   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.[54]

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[55] of two Banach spaces   and   is the completion   of the algebraic tensor product   equipped with the projective tensor norm, and similarly for the injective tensor product[56]   Grothendieck proved in particular that[57]

 

where   is a compact Hausdorff space,   the Banach space of continuous functions from   to   and   the space of Bochner-measurable and integrable functions from   to   and where the isomorphisms are isometric. The two isomorphisms above are the respective extensions of the map sending the tensor   to the vector-valued function  

Tensor products and the approximation propertyEdit

Let   be a Banach space. The tensor product   is identified isometrically with the closure in   of the set of finite rank operators. When   has the approximation property, this closure coincides with the space of compact operators on  

For every Banach space   there is a natural norm   linear map

 
obtained by extending the identity map of the algebraic tensor product. Grothendieck related the approximation problem to the question of whether this map is one-to-one when   is the dual of   Precisely, for every Banach space   the map
 
is one-to-one if and only if   has the approximation property.[58]

Grothendieck conjectured that   and   must be different whenever   and   are infinite-dimensional Banach spaces. This was disproved by Gilles Pisier in 1983.[59] Pisier constructed an infinite-dimensional Banach space   such that   and   are equal. Furthermore, just as Enflo's example, this space   is a "hand-made" space that fails to have the approximation property. On the other hand, Szankowski proved that the classical space   does not have the approximation property.[60]

Some classification resultsEdit

Characterizations of Hilbert space among Banach spacesEdit

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

for all  

It follows, for example, that the Lebesgue space   is a Hilbert space only when   If this identity is satisfied, the associated inner product is given by the polarization identity. In the case of real scalars, this gives:

 

For complex scalars, defining the inner product so as to be  -linear in   antilinear in   the polarization identity gives:

 

To see that the parallelogram law is sufficient, one observes in the real case that   is symmetric, and in the complex case, that it satisfies the Hermitian symmetry property and   The parallelogram law implies that   is additive in   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  : Kwapień proved that if

 
for every integer   and all families of vectors  then the Banach space X is isomorphic to a Hilbert space.[61] Here,   denotes the average over the   possible choices of signs   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.[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   any finite-dimensional normed space, with dimension sufficiently large compared to   contains subspaces nearly isometric to the  -dimensional Euclidean space.

The next result gives the solution of the so-called homogeneous space problem. An infinite-dimensional Banach space   is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces. A Banach space isomorphic to   is homogeneous, and Banach asked for the converse.[63]

Theorem.[64] A Banach space isomorphic to all its infinite-dimensional closed subspaces is isomorphic to a separable Hilbert space.

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[64] asserts that every infinite-dimensional Banach space X contains, either a subspace Y with unconditional basis, or a hereditarily indecomposable subspace Z, and in particular, Z is not isomorphic to its closed hyperplanes.[65] If X 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,[66] that X is isomorphic to  

Metric classificationEdit

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.

Topological classificationEdit

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.

Spaces of continuous functionsEdit

When two compact Hausdorff spaces   and   are homeomorphic, the Banach spaces   and   are isometric. Conversely, when   is not homeomorphic to   the (multiplicative) Banach–Mazur distance between   and   must be greater than or equal to 2, 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:[69]

Theorem.[70] Let K be an uncountable compact metric space. Then   is isomorphic to  

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

 

equipped with the order topology, where   is a countably infinite ordinal.[71] The Banach space   is then isometric to C(<1, α >). When   are two countably infinite ordinals, and assuming   the spaces C(<1, α >) and C(<1, β >) are isomorphic if and only if β < αω.[72] For example, the Banach spaces

 
are mutually non-isomorphic.

ExamplesEdit

Glossary of symbols for the table below:

  •   denotes the field of real numbers   or complex numbers  
  •   is a compact Hausdorff space.
  •   are real numbers with   that are Hölder conjugates, meaning that they satisfy   and thus also  
  •   is a  -algebra of sets.
  •   is an algebra of sets (for spaces only requiring finite additivity, such as the ba space).
  •   is a measure with variation   A positive measure is a real-valued positive set function defined on a  -algebra which is countably additive.
Classical Banach spaces
Dual space Reflexive weakly sequentially complete Norm Notes
    Yes Yes     Euclidean space
    Yes Yes    
    Yes Yes    
    Yes Yes    
    No Yes    
    No No    
    No No    
    No No     Isomorphic but not isometric to  
    No Yes     Isometrically isomorphic to  
    No Yes     Isometrically isomorphic to  
    No No     Isometrically isomorphic to  
    No No     Isometrically isomorphic to  
    No No    
    No No    
  ? No Yes    
  ? No Yes     A closed subspace of  
  ? No Yes     A closed subspace of  
    Yes Yes    
    No Yes     The dual is   if   is  -finite.
  ? No Yes       is the total variation of  
  ? No Yes       consists of   functions such that  
    No Yes     Isomorphic to the Sobolev space  
    No No     Isomorphic to   essentially by Taylor's theorem.

DerivativesEdit

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.

GeneralizationsEdit

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.

See alsoEdit

NotesEdit

  1. ^ It is common to read "  is a normed space" instead of the more technically correct but (usually) pedantic "  is a normed space," especially if the norm is well known (e.g. such as with Lp 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.
  2. ^ This means that if the norm   is replaced with a different norm   then   is not the same normed space as   even if the norms are equivalent. However, equivalence of norms on a given vector space does form an equivalence relation.
  3. ^ a b c A 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.
  4. ^ Because   for all   it is always true that   for all   So the order of   and   in this definition doesn't matter.
  5. ^ Whether or not a sequence is Cauchy in   depends on the metric   and not, say, just on the topology that   induces.
  6. ^ Indeed,   is even a locally convex metrizable topological vector space
  7. ^ Let   denote the Banach space of continuous functions with the supremum norm and let   denote the topology on   induced by   Since   can be embedded (via the canonical inclusion) as a vector subspace of   it is possible to define the restriction of the L1-norm to   which will be denoted by   This map   is a norm on   (in general, the restriction of any norm to any vector subspace will necessarily again be a norm). Because   the map   is continuous. However, the norm