where because this sequence's convergence to can equivalently be expressed as:
By definition, the normed space is a Banach space if the norm induced metric is a complete metric, or said differently, if is a complete metric space.
The norm of a normed space is called a complete norm if is a Banach space.
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. L-semi-inner products are a generalization of inner products, which are what fundamentally distinguish Hilbert spaces from all other Banach spaces. This shows that all normed spaces (and hence all Banach spaces) can be considered as being generalizations of (pre-)Hilbert spaces.
Characterization in terms of series
The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors.
A normed space is a Banach space if and only if each absolutely convergent series in converges in 
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
Using shows that this norm-induced topology is translation invariant, which means that for any and the subset is open (respectively, closed) in if and only if this is true of its translation
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 a sequence in of positive real numbers that converges to in (such as or for instance).
So for example, every open subset of can be written as a union
indexed by some subset where every is of the form for some integer (the closed ball can also be used instead of the open ball, although the indexing set and radii may need to be changed).
Additionally, can always be chosen to be countable if is a separable space, which by definition means that contains some countable dense subset.
The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space of countably many copies of (this homeomorphism need not be a linear map).
Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including the separable Hilbert2 sequence space with its usual norm where (in sharp contrast to finite−dimensional spaces) is also homeomorphic to its unit sphere
There is a compact subset of whose convex hull is not closed and thus also not compact (see this footnote[note 5] for an example).
However, like in all Banach spaces, the closed convex hull of this (and every other) compact subset will be compact. But if a normed space is not complete then it is in general not guaranteed that will be compact whenever is; an example[note 5] can even be found in a (non-complete) pre-Hilbert vector subspace of
This norm-induced topology also makes into what is known as a topological vector space (TVS), which by definition is a vector space endowed with a topology making the operations of addition and scalar multiplication continuous. It is emphasized that the TVS is 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"). 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 normable, which by definition refers to any TVS whose its topology is induced by some (possibly unknown) norm.
Comparison of complete metrizable vector topologies
The open mapping theorem implies that if are topologies on that make both and into complete metrizable TVS (for example, Banach or Fréchet spaces) and if one topology is finer or coarser than the other then they must be equal (that is, if ).
So for example, if are Banach spaces with topologies and if one of these spaces has some open ball that is also an open subset of the other space (or equivalently, if one of or is continuous) then their topologies are identical and their norms are equivalent.
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 positive 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 not equivalent to that Banach space's given norm.[note 6]
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[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 anymetric 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.
If is not translation invariant, then it may be possible for to be a Banach space but for to not be a complete metric space (see this footnote[note 7] for an example). In contrast, a theorem of Klee,[note 8] which also applies to all metrizable topological vector spaces, implies that if there exists any[note 9] 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 (that is, 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 (such as any norm induced topology, for example), 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).
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 Hausdorff 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 sometimes denoted by
If and are normed spaces over the same ground field the set of all continuous-linear maps is denoted by In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space to another normed space is continuous if and only if it is bounded on the closed unit ball of Thus, the vector space can be given the operator norm
For a Banach space, the space 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; in that case the space reappears as a natural bifunctor.
If is a Banach space, the space forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.
If and are normed spaces, they are isomorphic normed spaces if there exists a linear bijection such that and its inverse are continuous. If one of the two spaces or is complete (or reflexive, separable, etc.) then so is the other space. Two normed spaces and are isometrically isomorphic if in addition, is an isometry, that is, for every in The Banach–Mazur distance between two isomorphic but not isometric spaces and gives a measure of how much the two spaces and differ.
Continuous and bounded linear functions and seminormsEdit
The Cartesian product of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used, such as
which correspond (respectively) to the coproduct and product in the category of Banach spaces and short maps (discussed above). For finite (co)products, these norms give rise to isomorphic normed spaces, and the product (or the direct sum ) is complete if and only if the two factors are complete.
Suppose that and are Banach spaces and that There exists a canonical factorization of as
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.
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; 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  For every separable Banach space there is a closed subspace of such that 
Any Hilbert space serves as an example of a Banach space. A Hilbert space on is complete for a norm of the form
is the inner product, linear in its first argument that satisfies the following:
The Banach space 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: Equipped with the max norm on the disk algebra is a closed subalgebra of
The Wiener algebra is the algebra of functions on the unit circle with absolutely convergent Fourier series. Via the map associating a function on 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 the space of bounded linear operators on with the composition of maps as product, is a Banach algebra.
A C*-algebra is a complex Banach algebra with an antilinearinvolution such that The space of bounded linear operators on a Hilbert space 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 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
If 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 into or continuous linear functionals.
The notation for the continuous dual is in this article.
Since is a Banach space (using the absolute value as norm), the dual is a Banach space, for every normed space
The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem.
Hahn–Banach theorem — Let be a vector space over the field Let further
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.
An important special case is the following: for every vector in a normed space there exists a continuous linear functional on such that
When is not equal to the vector, the functional must have norm one, and is called a norming functional for
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 affinehyperplane.
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.
A subset in a Banach space is total if the linear span of is dense in The subset is total in if and only if the only continuous linear functional that vanishes on is the functional: this equivalence follows from the Hahn–Banach theorem.
If is the direct sum of two closed linear subspaces and then the dual of is isomorphic to the direct sum of the duals of and 
If is a closed linear subspace in one can associate the orthogonal of in the dual,
The orthogonal is a closed linear subspace of the dual. The dual of is isometrically isomorphic to
The dual of is isometrically isomorphic to 
The dual of a separable Banach space need not be separable, but:
Theorem — Let be a normed space. If is separable, then is separable.
When is separable, the above criterion for totality can be used for proving the existence of a countable total subset in
The weak topology on a Banach space is the coarsest topology on 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.
A norm-continuous linear map between two Banach spaces and is also weakly continuous, that is, continuous from the weak topology of to that of 
If is infinite-dimensional, there exist linear maps which are not continuous. The space of all linear maps from to the underlying field (this space is called the algebraic dual space, to distinguish it from also induces a topology on 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 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.
The Banach–Alaoglu theorem can be proved using Tychonoff's theorem about infinite products of compact Hausdorff spaces.
When is separable, the unit ball of the dual is a metrizable compact in the weak* topology.
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 is of the form for a uniquely defined vector in
The mapping is an antilinear isometric bijection from onto its dual
When the scalars are real, this map is an isometric isomorphism.
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
Theorem — If is a compact Hausdorff space, then the maximal ideal space of the Banach algebra is homeomorphic to 
Not every unital commutative Banach algebra is of the form for some compact Hausdorff space However, this statement holds if one places in the smaller category of commutative C*-algebras.
Gelfand'srepresentation theorem for commutative C*-algebras states that every commutative unital C*-algebra is isometrically isomorphic to a space.
The Hausdorff compact space here is again the maximal ideal space, also called the spectrum of in the C*-algebra context.
If is a normed space, the (continuous) dual of the dual is called bidual, or second dual of
For every normed space 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 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 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 as a subset of its bidual.
When is a Banach space, it is viewed as a closed linear subspace of If is not reflexive, the unit ball of 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 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.
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 be a Banach space and be a normed vector space. Suppose that is a collection of continuous linear operators from to The uniform boundedness principle states that if for all in we have then
The Banach–Steinhaus theorem is not limited to Banach spaces.
It can be extended for example to the case where is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood of in such that all in are uniformly bounded on
The Open Mapping Theorem — Let and be Banach spaces and be a surjective continuous linear operator, then 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 and are Banach spaces and that Suppose further that the range of is closed in 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 is the internal direct sum of closed subspaces then is isomorphic to
This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from onto sending to the sum
The Closed Graph Theorem — Let be a linear mapping between Banach spaces. The graph of is closed in if and only if is continuous.
The normed space is called reflexive when the natural map
is surjective. Reflexive normed spaces are Banach spaces.
Theorem — If is a reflexive Banach space, every closed subspace of and every quotient space of 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 onto the Banach space then is reflexive.
Theorem — If is a Banach space, then is reflexive if and only if is reflexive.
Corollary — Let be a reflexive Banach space. Then is separable if and only if is separable.
Indeed, if the dual of a Banach space is separable, then is separable.
If is reflexive and separable, then the dual of is separable, so is separable.
Theorem — Suppose that are normed spaces and that Then is reflexive if and only if each is reflexive.
Hilbert spaces are reflexive. The 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 the bidual is "much larger" than
Namely, under the natural isometric embedding of into given by the Hahn–Banach theorem, the quotient is infinite-dimensional, and even nonseparable.
However, Robert C. James has constructed an example of a non-reflexive space, usually called "the James space" and denoted by  such that the quotient is one-dimensional.
Furthermore, this space is isometrically isomorphic to its bidual.
When is reflexive, it follows that all closed and bounded convex subsets of are weakly compact.
In a Hilbert space the weak compactness of the unit ball is very often used in the following way: every bounded sequence in 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 of a reflexive space attains its minimum at some point in
As a special case of the preceding result, when is a reflexive space over every continuous linear functional in attains its maximum on the unit ball of
The following theorem of Robert C. James provides a converse statement.
James' Theorem — For a Banach space the following two properties are equivalent:
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 there exist continuous linear functionals that are not norm-attaining.
However, the Bishop–Phelps theorem states that norm-attaining functionals are norm dense in the dual of
A sequence in a Banach space is weakly convergent to a vector if converges to for every continuous linear functional in the dual The sequence is a weakly Cauchy sequence if converges to a scalar limit for every in
A sequence in the dual is weakly* convergent to a functional if converges to for every in
Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the Banach–Steinhaus theorem.
When the sequence in is a weakly Cauchy sequence, the limit above defines a bounded linear functional on the dual that is, an element of the bidual of and is the limit of in the weak*-topology of the bidual.
The Banach space is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent in
It follows from the preceding discussion that reflexive spaces are weakly sequentially complete.
Theorem  — 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 vector.
The unit vector basis of for or of is another example of a weakly null sequence, that is, a sequence that converges weakly to
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 
The unit vector basis of is not weakly Cauchy.
Weakly Cauchy sequences in are weakly convergent, since -spaces are weakly sequentially complete.
Actually, weakly convergent sequences in are norm convergent. This means that satisfies Schur's property.
Weakly Cauchy sequences and the basis are the opposite cases of the dichotomy established in the following deep result of H. P. Rosenthal.
Theorem — 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 — Let be a separable Banach space. The following are equivalent:
The space does not contain a closed subspace isomorphic to
Every element of the bidual is the weak*-limit of a sequence in
By the Goldstine theorem, every element of the unit ball of is weak*-limit of a net in the unit ball of When does not contain every element of is weak*-limit of a sequence in the unit ball of 
When the Banach space is separable, the unit ball of the dual equipped with the weak*-topology, is a metrizable compact space  and every element in the bidual defines a bounded function on :
This function is continuous for the compact topology of if and only if is actually in considered as subset of
Assume in addition for the rest of the paragraph that does not contain
By the preceding result of Odell and Rosenthal, the function is the pointwise limit on of a sequence of continuous functions on it is therefore a first Baire class function on
The unit ball of the bidual is a pointwise compact subset of the first Baire class on 
When 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.
The weak topology of a Banach space is metrizable if and only if is finite-dimensional. If the dual is separable, the weak topology of the unit ball of 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 — A set in a Banach space is relatively weakly compact if and only if every sequence in has a weakly convergent subsequence.
A Banach space is reflexive if and only if each bounded sequence in has a weakly convergent subsequence.
A weakly compact subset in is norm-compact. Indeed, every sequence in has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of
A Schauder basis in a Banach space is a sequence of vectors in with the property that for every vector there exist uniquely defined scalars depending on 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 are uniformly bounded by some constant
Let denote the coordinate functionals which assign to every in the coordinate of in the above expansion.
They are called biorthogonal functionals. When the basis vectors have norm the coordinate functionals have norm in the dual of
Most classical separable spaces have explicit bases.
The Haar system is a basis for
The trigonometric system is a basis in when
The Schauder system is a basis in the space 
The question of whether the disk algebra has a basis remained open for more than forty years, until Bočkarev showed in 1974 that admits a basis constructed from the Franklin system.
Since every vector in a Banach space with a basis is the limit of with of finite rank and uniformly bounded, the space 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.
Robert C. James characterized reflexivity in Banach spaces with a basis: the space with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete.
In this case, the biorthogonal functionals form a basis of the dual of
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 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
Grothendieck proved in particular that
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.
Grothendieck conjectured that and must be different whenever and are infinite-dimensional Banach spaces.
This was disproved by Gilles Pisier in 1983.
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.
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:
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 is isomorphic to a Hilbert space.
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. 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.
Theorem — 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 asserts that every infinite-dimensional Banach space contains, either a subspace with unconditional basis, or a hereditarily indecomposable subspace and in particular, is not isomorphic to its closed hyperplanes.
If 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, that is isomorphic to
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.
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.
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 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:
Theorem — Let be an uncountable compact metric space. Then is isomorphic to
The situation is different for countably infinite compact Hausdorff spaces.
Every countably infinite compact is homeomorphic to some closed interval of ordinal numbers
equipped with the order topology, where is a countably infinite ordinal.
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.
For example, the Banach spaces
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 convextopological 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.
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.