# Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced Template:IPA-pl) 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 in the space.

Banach spaces are named after the Polish mathematician Stefan Banach, who introduced and made a systematic study of them in 1920–1922 along with Hans Hahn and Eduard Helly.[1] Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces.

## Definition

A Banach space is a vector space Template:Mvar over the field R of real numbers, or over the field C of complex numbers, which is equipped with a norm and which is complete with respect to that norm, that is to say, for every Cauchy sequence {xn} in Template:Mvar, there exists an element Template:Mvar in Template:Mvar such that

${\displaystyle \lim _{n\to \infty }x_{n}=x,}$

or equivalently:

${\displaystyle \lim _{n\to \infty }\left\|x_{n}-x\right\|_{X}=0.}$

The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space Template:Mvar is a Banach space if and only if each absolutely convergent series in Template:Mvar converges,[2]

${\displaystyle \sum _{n=1}^{\infty }\|v_{n}\|_{X}<\infty \quad {\text{implies that}}\quad \sum _{n=1}^{\infty }v_{n}\ \ {\text{converges in}}\ \ X.}$

Completeness of a normed space is preserved if the given norm is replaced by an equivalent one.

All norms on a finite-dimensional vector space are equivalent. Every finite-dimensional normed space over R or C is a Banach space.[3]

## General theory

### Linear operators, isomorphisms

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

${\displaystyle \|T\|=\sup \left\{\|Tx\|_{Y}\mid x\in X,\ \|x\|_{X}\leq 1\right\}.}$

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

If Template:Mvar is a Banach space, the space B(X) = B(X, X) forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.

If Template:Mvar and Template:Mvar are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T : XY such that Template:Mvar and its inverse T −1 are continuous. If one of the two spaces Template:Mvar or Template:Mvar is complete (or reflexive, separable, etc.) then so is the other space. Two normed spaces Template:Mvar and Template:Mvar are isometrically isomorphic if in addition, Template:Mvar is an isometry, i.e., for every Template:Mvar in Template:Mvar. The Banach-Mazur distance d(X, Y) between two isomorphic but not isometric spaces Template:Mvar and Template:Mvar gives a measure of how much the two spaces Template:Mvar and Template:Mvar differ.

### Basic notions

Every normed space Template:Mvar can be isometrically embedded in a Banach space. More precisely, there is a Banach space Template:Mvar and an isometric mapping T : XY such that T(X) is dense in Template:Mvar. If Template:Mvar is another Banach space such that there is an isometric isomorphism from Template:Mvar onto a dense subset of Template:Mvar, then Template:Mvar is isometrically isomorphic to Template:Mvar.

This Banach space Template:Mvar is the completion of the normed space Template:Mvar. The underlying metric space for Template:Mvar is the same as the metric completion of Template:Mvar, with the vector space operations extended from Template:Mvar to Template:Mvar. The completion of Template:Mvar is often denoted by ${\displaystyle {\widehat {X}}}$.

The cartesian product X × Y of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used,[4] such as

${\displaystyle \|(x,y)\|_{1}=\|x\|+\|y\|,\qquad \|(x,y)\|_{\infty }=\max(\|x\|,\|y\|)}$

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

If Template:Mvar is a closed linear subspace of a normed space Template:Mvar, there is a natural norm on the quotient space X / M,

${\displaystyle \|x+M\|=\inf \limits _{m\in M}\|x+m\|.}$

The quotient X / M is a Banach space when Template:Mvar is complete.[5] The quotient map from Template:Mvar onto X / M, sending Template:Mvar in Template:Mvar to its class x + M, is linear, onto and has norm 1, except when M = X, in which case the quotient is the null space.

The closed linear subspace Template:Mvar of Template:Mvar is said to be a complemented subspace of Template:Mvar if Template:Mvar is the range of a bounded linear projection Template:Mvar from Template:Mvar onto Template:Mvar. In this case, the space Template:Mvar is isomorphic to the direct sum of Template:Mvar and Ker(P), the kernel of the projection Template:Mvar.

Suppose that Template:Mvar and Template:Mvar are Banach spaces and that TB(X, Y). There exists a canonical factorization of Template:Mvar as[5]

${\displaystyle T=T_{1}\circ \pi ,\ \ \ T:X\ {\overset {\pi }{\longrightarrow }}\ X/\operatorname {Ker} (T)\ {\overset {T_{1}}{\longrightarrow }}\ Y}$

where the first map Template:Mvar is the quotient map, and the second map T1 sends every class x + Ker(T) in the quotient to the image T(x) in Template:Mvar. This is well defined because all elements in the same class have the same image. The mapping T1 is a linear bijection from X / Ker(T) onto the range T(X), whose inverse need not be bounded.

### Classical spaces

Basic examples[6] of Banach spaces include: the Lp spaces and their special cases, the sequence spaces p that consist of scalar sequences indexed by N; among them, the space 1 of absolutely summable sequences and the space 2 of square summable sequences; the space c0 of sequences tending to zero and the space of bounded sequences; the space C(K) of continuous scalar functions on a compact Hausdorff space Template:Mvar, equipped with the max norm,

${\displaystyle \|f\|_{C(K)}=\max\{|f(x)|:x\in K\},\quad f\in C(K).}$

According to the Banach–Mazur theorem, every Banach space is isometrically isomorphic to a subspace of some C(K).[7] For every separable Banach space Template:Mvar, there is a closed subspace Template:Mvar of 1 such that X ≅ ℓ1/M.[8]

Any Hilbert space serves as an example of a Banach space. A Hilbert space Template:Mvar on K = R, C is complete for a norm of the form

${\displaystyle \|x\|_{H}={\sqrt {\langle x,x\rangle }},}$

where

${\displaystyle \langle \cdot ,\cdot \rangle :H\times H\to \mathbf {K} }$

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

{\displaystyle {\begin{aligned}\forall x,y\in H:\quad \langle y,x\rangle &={\overline {\langle x,y\rangle }},\\\forall x\in H:\quad \langle x,x\rangle &\geq 0,\\\langle x,x\rangle =0\Leftrightarrow x&=0.\end{aligned}}}

For example, the space L2 is a Hilbert space.

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.

### Banach algebras

A Banach algebra is a Banach space Template:Mvar over K = R or C, together with a structure of algebra over K, such that the product map (a, b) ∈ A × AabA is continuous. An equivalent norm on Template:Mvar can be found so that for all a, bA.

#### Examples

• The Banach space C(K), with the pointwise product, is a Banach algebra.
• The disk algebra A(D) consists of functions holomorphic in the open unit disk DC and continuous on its closure: Template:Overline. Equipped with the max norm on Template:Overline, the disk algebra A(D) is a closed subalgebra of .
• 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 1(Z), where the product is the convolution of sequences.
• For every Banach space Template:Mvar, the space B(X) of bounded linear operators on Template:Mvar, with the composition of maps as product, is a Banach algebra.
• A C*-algebra is a complex Banach algebra Template:Mvar with an antilinear involution aa such that . The space B(H) of bounded linear operators on a Hilbert space Template:Mvar 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 C(K) of complex continuous functions on a compact Hausdorff space Template:Mvar is an example of commutative C*-algebra, where the involution associates to every function f its complex conjugate Template:Overline.

### Dual space

{{#invoke:main|main}} If Template:Mvar is a normed space and K the underlying field (either the real or the complex numbers), the continuous dual space is the space of continuous linear maps from Template:Mvar into K, or continuous linear functionals. The notation for the continuous dual is X ′ = B(X, K) in this article.[9] Since K is a Banach space (using the absolute value as norm), the dual X ′ is a Banach space, for every normed space Template:Mvar.

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

Hahn–Banach theorem. Let Template:Mvar be a vector space over the field K = R, C. Let further
Then, there exists a linear functional F : XK so that
${\displaystyle F|_{Y}=f,\quad {\text{and}}\quad \forall x\in X,\ \ \operatorname {Re} (F(x))\leq p(x).}$

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.[10] An important special case is the following: for every vector Template:Mvar in a normed space Template:Mvar, there exists a continuous linear functional f on Template:Mvar such that

${\displaystyle f(x)=\|x\|_{X},\quad \|f\|_{X'}\leq 1.}$

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

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.[11]

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

If Template:Mvar is the direct sum of two closed linear subspaces Template:Mvar and Template:Mvar, then the dual X ′ of Template:Mvar is isomorphic to the direct sum of the duals of Template:Mvar and Template:Mvar.[12] If Template:Mvar is a closed linear subspace in Template:Mvar, one can associate the orthogonal of Template:Mvar in the dual,

${\displaystyle M^{\perp }=\left\{x'\in X':x'(m)=0,\ \forall m\in M\right\}.}$

The orthogonal M ⊥ is a closed linear subspace of the dual. The dual of Template:Mvar is isometrically isomorphic to X ′ / M ⊥. The dual of X / M is isometrically isomorphic to M ⊥.[13]

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

Theorem.[14] Let Template:Mvar be a normed space. If X ′ is separable, then Template:Mvar is separable.

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

#### Weak topologies

The weak topology on a Banach space Template:Mvar is the coarsest topology on Template:Mvar for which all elements x ′ in the continuous dual space X ′ 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.[15] A norm-continuous linear map between two Banach spaces Template:Mvar and Template:Mvar is also weakly continuous, i.e., continuous from the weak topology of Template:Mvar to that of Template:Mvar.[16]

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

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

Banach–Alaoglu Theorem. Let Template:Mvar be a normed vector space. Then the closed unit ball B ′ = {x′ ∈ X ′ : Template:!!xTemplate:!! ≤ 1} of the dual space is compact in the weak* topology.

The Banach–Alaoglu theorem depends on Tychonoff's theorem about infinite products of compact spaces. When Template:Mvar is separable, the unit ball B ′ of the dual is a metrizable compact in the weak* topology.[17]

#### Examples of dual spaces

The dual of c0 is isometrically isomorphic to 1: for every bounded linear functional f on c0, there is a unique element y = {yn} ∈ ℓ1 such that

${\displaystyle f(x)=\sum _{n\in \mathbf {N} }x_{n}y_{n},\qquad x=\{x_{n}\}\in c_{0},\ \ {\text{and}}\ \ \|f\|_{(c_{0})'}=\|y\|_{\ell _{1}}.}$

The dual of 1 is isometrically isomorphic to . The dual of Lp([0, 1]) is isometrically isomorphic to Lq([0, 1]) when 1 ≤ p < ∞ and  {{ safesubst:#invoke:Unsubst||$B=1/p}} + {{ safesubst:#invoke:Unsubst||$B=1/q}} = 1.

For every vector Template:Mvar in a Hilbert space Template:Mvar, the mapping

${\displaystyle x\in H\to f_{y}(x)=\langle x,y\rangle }$

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

When Template:Mvar is a compact Hausdorff topological space, the dual M(K) of C(K) is the space of Radon measures in the sense of Bourbaki.[18] 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 Template:Mvar. The set of Dirac measures on Template:Mvar, equipped with the w*-topology, is homeomorphic to Template:Mvar.

Banach-Stone Theorem. If Template:Mvar and Template:Mvar are compact Hausdorff spaces and if C(K) and C(L) are isometrically isomorphic, then the topological spaces Template:Mvar and Template:Mvar are homeomorphic.[19][20]

The result has been extended by Amir[21] and Cambern[22] 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.[23]

In the commutative Banach algebra C(K), the maximal ideals are precisely kernels of Dirac mesures on Template:Mvar,

${\displaystyle I_{x}=\ker \delta _{x}=\{f\in C(K):f(x)=0\},\quad x\in K.}$

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 Template:Mvar is a compact Hausdorff space, then the maximal ideal space Ξ of the Banach algebra C(K) is homeomorphic to Template:Mvar.[19]

Not every unital commutative Banach algebra is of the form C(K) for some compact Hausdorff space Template:Mvar. 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 Template:Mvar is isometrically isomorphic to a C(K) space.[24] The Hausdorff compact space Template:Mvar here is again the maximal ideal space, also called the spectrum of Template:Mvar in the C*-algebra context.

#### Bidual

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

${\displaystyle {\begin{cases}F_{X}:X\to X''\\F_{X}(x)(f)=f(x)&\forall x\in X,\forall f\in X'\end{cases}}}$

This defines FX(x) as a continuous linear functional on X ′, i.e., an element of X ′′. The map FX : xFX(x) is a linear map from Template:Mvar to X ′′. As a consequence of the existence of a norming functional f for every Template:Mvar in Template:Mvar, this map FX is isometric, thus injective.

For example, the dual of X = c0 is identified with 1, and the dual of 1 is identified with , the space of bounded scalar sequences. Under these identifications, FX is the inclusion map from c0 to . It is indeed isometric, but not onto.

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

Using the isometric embedding FX, it is customary to consider a normed space Template:Mvar as a subset of its bidual. When Template:Mvar is a Banach space, it is viewed as a closed linear subspace of X ′′. If Template:Mvar is not reflexive, the unit ball of Template:Mvar is a proper subset of the unit ball of X ′′. 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 x ′′ in the bidual, there exists a net {xj} in Template:Mvar so that

${\displaystyle \sup _{j}\|x_{j}\|\leq \|x''\|,\ \ x''(f)=\lim _{j}f(x_{j}),\quad f\in X'.}$

The net may be replaced by a weakly*-convergent sequence when the dual X ′ is separable. On the other hand, elements of the bidual of 1 that are not in 1 cannot be weak*-limit of sequences in 1, since 1 is weakly sequentially complete.

### Banach's theorems

Here are the main general results about Banach spaces that go back to the time of Banach's book (Template:Harvtxt) 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 Template:Mvar be a Banach space and Template:Mvar be a normed vector space. Suppose that Template:Mvar is a collection of continuous linear operators from Template:Mvar to Template:Mvar. The uniform boundedness principle states that if for all Template:Mvar in Template:Mvar we have supTF Template:!!T(x)Template:!!Y < ∞, then supTF Template:!!TTemplate:!!Y < ∞.

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

${\displaystyle \sup _{T\in F}\sup _{x\in U}\;\|T(x)\|_{Y}<\infty .}$
The Open Mapping Theorem. Let Template:Mvar and Template:Mvar be Banach spaces and T : XY be a continuous linear operator. Then Template:Mvar is surjective if and only if Template:Mvar 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 Template:Mvar and Template:Mvar are Banach spaces and that TB(X, Y). Suppose further that the range of Template:Mvar is closed in Template:Mvar. Then X/ Ker(T) is isomorphic to T(X).

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 Template:Mvar is the internal direct sum of closed subspaces M1, ..., Mn, then Template:Mvar is isomorphic to M1 ⊕ ... ⊕ Mn.

This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from M1 ⊕ ... ⊕ Mn onto Template:Mvar sending (m1, ..., mn) to the sum m1 + ... + mn.

The Closed Graph Theorem. Let T : XY be a linear mapping between Banach spaces. The graph of Template:Mvar is closed in X × Y if and only if Template:Mvar is continuous.

### Reflexivity

{{#invoke:main|main}} The normed space Template:Mvar is called reflexive when the natural map

${\displaystyle {\begin{cases}F_{X}:X\to X''\\F_{X}(x)(f)=f(x)&\forall x\in X,\forall f\in X'\end{cases}}}$

is surjective. Reflexive normed spaces are Banach spaces.

Theorem. If Template:Mvar is a reflexive Banach space, every closed subspace of Template:Mvar and every quotient space of Template:Mvar 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 Template:Mvar onto the Banach space Template:Mvar, then Template:Mvar is reflexive.

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

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

Theorem. Suppose that X1, ..., Xn are normed spaces and that X = X1 ⊕ ... ⊕ Xn. Then Template:Mvar is reflexive if and only if each Xj is reflexive.

Hilbert spaces are reflexive. The Lp spaces are reflexive when 1 < p < ∞. More generally, uniformly convex spaces are reflexive, by the Milman–Pettis theorem. The spaces c0, ℓ1, L1([0, 1]), C([0, 1]) are not reflexive. In these examples of non-reflexive spaces Template:Mvar, the bidual X ′′ is "much larger" than Template:Mvar. Namely, under the natural isometric embedding of Template:Mvar 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[25] of a non-reflexive space, usually called "the James space" and denoted by J,[26] such that the quotient J ′′ / J is one-dimensional. Furthermore, this space Template:Mvar is isometrically isomorphic to its bidual.

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

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

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

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 Template:Mvar, there exist continuous linear functionals that are not norm-attaining. However, the BishopPhelps theorem[27] states that norm-attaining functionals are norm dense in the dual X ′ of Template:Mvar.

### Weak convergences of sequences

A sequence {xn} in a Banach space Template:Mvar is weakly convergent to a vector xX if f (xn) converges to f (x) for every continuous linear functional f in the dual X ′. The sequence {xn} is a weakly Cauchy sequence if f (xn) converges to a scalar limit L( f ), for every f in X ′. A sequence { fn } in the dual X ′ is weakly* convergent to a functional f  ∈ X ′ if fn (x) converges to f (x) for every Template:Mvar in Template:Mvar. Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the Banach–Steinhaus theorem.

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

Theorem. [28] For every measure Template:Mvar, the space L1(μ) 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 p, 1 < p < ∞, or of c0, 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.[29]

The unit vector basis of 1 is not weakly Cauchy. Weakly Cauchy sequences in 1 are weakly convergent, since L1-spaces are weakly sequentially complete. Actually, weakly convergent sequences in 1 are norm convergent.[30] This means that 1 satisfies Schur's property.

#### Results involving the ℓ1 basis

Weakly Cauchy sequences and the 1 basis are the opposite cases of the dichotomy established in the following deep result of H. P. Rosenthal.[31]

Theorem.[32] Let {xn} be a bounded sequence in a Banach space. Either {xn} has a weakly Cauchy subsequence, or it admits a subsequence equivalent to the standard unit vector basis of 1.

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

Theorem.[33] Let Template:Mvar be a separable Banach space. The following are equivalent:
• The space Template:Mvar does not contain a closed subspace isomorphic to 1.
• Every element of the bidual X ′′ is the weak*-limit of a sequence {xn} in Template:Mvar.

By the Goldstine theorem, every element of the unit ball B ′′ of X ′′ is weak*-limit of a net in the unit ball of Template:Mvar. When Template:Mvar does not contain 1, every element of B ′′ is weak*-limit of a sequence in the unit ball of Template:Mvar.[34]

When the Banach space Template:Mvar is separable, the unit ball of the dual X ′, equipped with the weak*-topology, is a metrizable compact space Template:Mvar,[17] and every element x ′′ in the bidual X ′′ defines a bounded function on Template:Mvar:

${\displaystyle x'\in K\mapsto x''(x'),\quad \left|x''(x')\right|\leq \left\|x''\right\|.}$

This function is continuous for the compact topology of Template:Mvar if and only if x ′′ is actually in Template:Mvar, considered as subset of X ′′. Assume in addition for the rest of the paragraph that Template:Mvar does not contain 1. By the preceding result of Odell and Rosenthal, the function x ′′ is the pointwise limit on Template:Mvar of a sequence {xn} ⊂ X of continuous functions on Template:Mvar, it is therefore a first Baire class function on Template:Mvar. The unit ball of the bidual is a pointwise compact subset of the first Baire class on Template:Mvar.[35]

#### Sequences, weak and weak* compactness

When Template:Mvar is separable, the unit ball of the dual is weak*-compact by Banach–Alaoglu and metrizable for the weak* topology,[17] 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 Template:Mvar is metrizable if and only if Template:Mvar is finite-dimensional.[36] If the dual X ′ is separable, the weak topology of the unit ball of Template:Mvar 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.[37] A set Template:Mvar in a Banach space is relatively weakly compact if and only if every sequence {an} in Template:Mvar has a weakly convergent subsequence.

A Banach space Template:Mvar is reflexive if and only if each bounded sequence in Template:Mvar has a weakly convergent subsequence.[38]

A weakly compact subset Template:Mvar in 1 is norm-compact. Indeed, every sequence in Template:Mvar has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of 1.

## Schauder bases

{{#invoke:main|main}} A Schauder basis in a Banach space Template:Mvar is a sequence {en}n ≥ 0 of vectors in X with the property that for every vector Template:Mvar in Template:Mvar, there exist uniquely defined scalars {xn}n ≥ 0 depending on Template:Mvar, such that

${\displaystyle x=\sum _{n=0}^{\infty }x_{n}e_{n},\quad {\textit {i.e.,}}\quad x=\lim _{n}P_{n}(x),\ P_{n}(x):=\sum _{k=0}^{n}x_{k}e_{k}.}$

It follows from the Banach–Steinhaus theorem that the linear mappings {Pn} are uniformly bounded by some constant Template:Mvar. Let {eTemplate:Su} denote the coordinate functionals which assign to every Template:Mvar in Template:Mvar the coordinate xn of Template:Mvar in the above expansion. They are called biorthogonal functionals. When the basis vectors have norm 1, the coordinate functionals {eTemplate:Su} have norm ≤ 2C in the dual of Template:Mvar.

Most classical spaces have explicit bases. The Haar system {hn} is a basis for Lp([0, 1]), 1 ≤ p < ∞. The trigonometric system is a basis in Lp(T) when 1 < p < ∞. The Schauder system is a basis in the space C([0, 1]).[39] The question of whether the disk algebra A(D) has a basis[40] remained open for more than forty years, until Bočkarev showed in 1974 that A(D) admits a basis constructed from the Franklin system.[41]

Since every vector Template:Mvar in a Banach space Template:Mvar with a basis is the limit of Pn(x), with Pn of finite rank and uniformly bounded, the space Template:Mvar satisfies the bounded approximation property. The first example[42] by Enflo of a space failing the approximation property was at the same time the first example of a Banach space without Schauder basis.

Robert C. James characterized reflexivity in Banach spaces with basis: the space Template:Mvar with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete.[43] In this case, the biorthogonal functionals form a basis of the dual of Template:Mvar.

## Tensor product

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Let Template:Mvar and Template:Mvar be two K-vector spaces. The tensor product XY of Template:Mvar and Template:Mvar is a K-vector space Template:Mvar with a bilinear mapping T : X × YZ which has the following universal property:

If T1 : X × YZ1 is any bilinear mapping into a K-vector space Z1, then there exists a unique linear mapping f  : ZZ1 such that T1 = fT.

The image under Template:Mvar of a couple (x, y) in X × Y is denoted by xy, and called a simple tensor. Every element Template:Mvar in XY 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.[44]

In general, the tensor product of complete spaces is not complete again. When working with Banach spaces, it is customary to call projective tensor product[45] of two Banach spaces Template:Mvar and Template:Mvar the completion ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ of the algebraic tensor product XY equipped with the projective tensor norm, and similarly for the injective tensor product[46] ${\displaystyle X{\widehat {\otimes }}_{\varepsilon }Y}$. Grothendieck proved in particular that[47]

{\displaystyle {\begin{aligned}C(K){\widehat {\otimes }}_{\varepsilon }Y&\simeq C(K,Y),\\L^{1}([0,1]){\widehat {\otimes }}_{\pi }Y&\simeq L^{1}([0,1],Y),\end{aligned}}}

where Template:Mvar is a compact Hausdorff space, C(K, Y) the Banach space of continuous functions from Template:Mvar to Template:Mvar and L1([0, 1], Y) the space of Bochner-measurable and integrable functions from [0, 1] to Template:Mvar, and where the isomorphisms are isometric. The two isomorphisms above are the respective extensions of the map sending the tensor f  ⊗ y to the vector-valued function sK →  f (s)yY.

### Tensor products and the approximation property

Let Template:Mvar be a Banach space. The tensor product ${\displaystyle X'{\widehat {\otimes }}_{\varepsilon }X}$ is identified isometrically with the closure in B(X) of the set of finite rank operators. When Template:Mvar has the approximation property, this closure coincides with the space of compact operators on Template:Mvar.

For every Banach space Template:Mvar, there is a natural norm 1 linear map

${\displaystyle Y{\widehat {\otimes }}_{\pi }X\to Y{\widehat {\otimes }}_{\varepsilon }X}$

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 Template:Mvar is the dual of Template:Mvar. Precisely, for every Banach space Template:Mvar, the map

${\displaystyle X'{\widehat {\otimes }}_{\pi }X\ \longrightarrow X'{\widehat {\otimes }}_{\varepsilon }X}$

is one-to-one if and only if Template:Mvar has the approximation property.[48]

Grothendieck conjectured that ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ and ${\displaystyle X{\widehat {\otimes }}_{\varepsilon }Y}$ must be different whenever Template:Mvar and Template:Mvar are infinite-dimensional Banach spaces. This was disproved by Gilles Pisier in 1983.[49] Pisier constructed an infinite-dimensional Banach space Template:Mvar such that ${\displaystyle X{\widehat {\otimes }}_{\pi }X}$ and ${\displaystyle X{\widehat {\otimes }}_{\varepsilon }X}$ are equal. Furthermore, just as Enflo's example, this space Template:Mvar is a "hand-made" space that fails to have the approximation property. On the other hand, Szankowski proved that the classical space B(ℓ2) does not have the approximation property.[50]

## Some classification results

### Characterizations of Hilbert space among Banach spaces

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

${\displaystyle \forall x,y\in X:\qquad \|x+y\|^{2}+\|x-y\|^{2}=2\left(\|x\|^{2}+\|y\|^{2}\right).}$

It follows, for example, that the Lebesgue space Lp([0, 1]) is a Hilbert space only when p = 2. If this identity is satisfied, the associated inner product is given by the polarization identity. In the case of real scalars, this gives:

${\displaystyle \langle x,y\rangle ={\tfrac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}\right).}$

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

${\displaystyle \langle x,y\rangle ={\tfrac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}+i\left(\|x+iy\|^{2}-\|x-iy\|^{2}\right)\right).}$

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 Template:Mvar. 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

${\displaystyle c^{-2}\sum _{k=1}^{n}\left\|x_{k}\right\|^{2}\leq \operatorname {Ave} _{\pm }\left\|\sum _{k=1}^{n}\pm x_{k}\right\|^{2}\leq c^{2}\sum _{k=1}^{n}\left\|x_{k}\right\|^{2}}$

for every integer Template:Mvar and all families of vectors {x1, ..., xn} ⊂ X, then the Banach space Template:Mvar is isomorphic to a Hilbert space.[51] Here, Ave± denotes the average over the 2n possible choices of signs ±1. 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.[52] 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 Template:Mvar, any finite-dimensional normed space, with dimension sufficiently large compared to Template:Mvar, contains subspaces nearly isometric to the Template:Mvar-dimensional Euclidean space.

The next result gives the solution of the so-called homogeneous space problem. An infinite-dimensional Banach space Template:Mvar is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces. A Banach space isomorphic to 2 is homogeneous, and Banach asked for the converse.[53]

Theorem.[54] 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[54] asserts that every infinite-dimensional Banach space Template:Mvar contains, either a subspace Template:Mvar with unconditional basis, or a hereditarily indecomposable subspace Template:Mvar, and in particular, Template:Mvar is not isomorphic to its closed hyperplanes.[55] If Template:Mvar 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,[56] that Template:Mvar is isomorphic to 2.

### Spaces of continuous functions

When two compact Hausdorff spaces K1 and K2 are homeomorphic, the Banach spaces C(K1) and C(K2) are isometric. Conversely, when K1 is not homeomorphic to K2, the (multiplicative) Banach–Mazur distance between C(K1) and C(K2) 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:[57]

Theorem.[58] Let Template:Mvar be an uncountable compact metric space. Then C(K) is isomorphic to C([0, 1]).

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

${\displaystyle \langle 1,\alpha \rangle =\{\gamma \ :\ 1\leq \gamma \leq \alpha \}}$

equipped with the order topology, where α is a countably infinite ordinal.[59] The Banach space C(K) is then isometric to C(<1, α >). When α, β are two countably infinite ordinals, and assuming αβ, the spaces C(<1, α >) and C(<1, β >) are isomorphic if and only if β < αω.[60] For example, the Banach spaces

${\displaystyle C(\langle 1,\omega \rangle ),\ C(\langle 1,\omega ^{\omega }\rangle ),\ C(\langle 1,\omega ^{\omega ^{2}}\rangle ),\ C(\langle 1,\omega ^{\omega ^{3}}\rangle ),\cdots ,C(\langle 1,\omega ^{\omega ^{\omega }}\rangle ),\cdots }$

are mutually non-isomorphic.

## Examples

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A glossary of symbols:

 Dual space Reflexive weakly sequentially complete Norm Classical Banach spaces Kn Yes Yes ${\displaystyle \|x\|_{2}=\left(\sum _{i=1}^{n}|x_{i}|^{2}\right)^{\frac {1}{2}}}$ Euclidean space Yes Yes ${\displaystyle \|x\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{\frac {1}{p}}}$ Yes Yes ${\displaystyle \|x\|_{\infty }=\max \nolimits _{1\leq i\leq n}|x_{i}|}$ ℓq Yes Yes ${\displaystyle \|x\|_{p}=\left(\sum _{i=1}^{\infty }|x_{i}|^{p}\right)^{\frac {1}{p}}}$ ℓ∞ No Yes ${\displaystyle \|x\|_{1}=\sum _{i=1}^{\infty }|x_{i}|}$ ba No No ${\displaystyle \|x\|_{\infty }=\sup \nolimits _{i}|x_{i}|}$ ℓ1 No No ${\displaystyle \|x\|_{\infty }=\sup \nolimits _{i}|x_{i}|}$ ℓ1 No No ${\displaystyle \|x\|_{\infty }=\sup \nolimits _{i}|x_{i}|}$ Isomorphic but not isometric to Template:Mvar. ℓ∞ No Yes ${\displaystyle \|x\|_{bv}=|x_{1}|+\sum _{i=1}^{\infty }|x_{i+1}-x_{i}|}$ Isometrically isomorphic to ℓ1. ℓ∞ No Yes ${\displaystyle \|x\|_{bv_{0}}=\sum _{i=1}^{\infty }|x_{i+1}-x_{i}|}$ Isometrically isomorphic to ℓ1. ba No No ${\displaystyle \|x\|_{bs}=\sup \nolimits _{n}\left|\sum _{i=1}^{n}x_{i}\right|}$ Isometrically isomorphic to ℓ∞. ℓ1 No No ${\displaystyle \|x\|_{bs}=\sup \nolimits _{n}\left|\sum _{i=1}^{n}x_{i}\right|}$ Isometrically isomorphic to [[c space|Template:Mvar]]. ba(Ξ) No No ${\displaystyle \|f\|_{B}=\sup \nolimits _{x\in X}|f(x)|}$ rca(X) No No ${\displaystyle \|x\|_{C(X)}=\max \nolimits _{x\in X}|f(x)|}$ ? No Yes ${\displaystyle \|\mu \|_{ba}=\sup \nolimits _{A\in \Sigma }|\mu |(A)}$ ? No Yes ${\displaystyle \|\mu \|_{ba}=\sup \nolimits _{A\in \Sigma }|\mu |(A)}$ A closed subspace of ba(Σ). ? No Yes ${\displaystyle \|\mu \|_{ba}=\sup \nolimits _{A\in \Sigma }|\mu |(A)}$ A closed subspace of ca(Σ). Lq(μ) Yes Yes ${\displaystyle \|f\|_{p}=\left(\int |f|^{p}\,d\mu \right)^{\frac {1}{p}}}$ L∞(μ) No Yes ${\displaystyle \|f\|_{1}=\int |f|\,d\mu }$ The dual is L∞(μ) if Template:Mvar is [[σ-finite measure|Template:Mvar-finite]]. ? No Yes ${\displaystyle \|f\|_{BV}=V_{f}(I)+\lim \nolimits _{x\to a^{+}}f(x)}$ Vf (I) is the total variation of  f ? No Yes ${\displaystyle \|f\|_{BV}=V_{f}(I)}$ NBV(I) consists of BV(I) functions such that ${\displaystyle \lim \nolimits _{x\to a^{+}}f(x)=0}$ K + L∞(I) No Yes ${\displaystyle \|f\|_{BV}=V_{f}(I)+\lim \nolimits _{x\to a^{+}}f(x)}$ Isomorphic to the Sobolev space W 1,1(I). rca([a,b]) No No ${\displaystyle \|f\|=\sum _{i=0}^{n}\sup \nolimits _{x\in [a,b]}\left|f^{(i)}(x)\right|}$ Isomorphic to Rn ⊕ C([a,b]), essentially by Taylor's theorem.

## Derivatives

Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gâteaux derivative for details. The Fréchet derivative allows for an extension of the concept of a directional derivative to Banach spaces. The Gâteaux derivative allows for an extension of a directional derivative to locally convex topological vector spaces. Fréchet differentiability is a stronger condition than Gâteaux differentiability. The quasi-derivative is another generalization of directional derivative that implies a stronger condition than Gâteaux differentiability, but a weaker condition than Fréchet differentiability.

## Generalizations

Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions RR, or the space of all distributions on R, 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.

## Notes

1. Template:Harvnb
2. see Theorem 1.3.9, p. 20 in Template:Harvtxt.
3. see Corollary 1.4.18, p. 32 in Template:Harvtxt.
4. see Template:Harvtxt, p. 182.
5. see pp. 17–19 in Template:Harvtxt.
6. see Template:Harvtxt, pp. 11-12.
7. see Template:Harvtxt, Th. 9 p. 185.
8. see Theorem 6.1, p. 55 in Template:Harvtxt
9. Several books about functional analysis use the notation X ∗ for the continuous dual, for example Template:Harvtxt, Template:Harvtxt, Template:Harvtxt, Template:Harvtxt, Template:Harvtxt.
10. Theorem 1.9.6, p. 75 in Template:Harvtxt
12. see p. 19 in Template:Harvtxt.
13. Theorems 1.10.16, 1.10.17 pp.94–95 in Template:Harvtxt
14. Theorem 1.12.11, p. 112 in Template:Harvtxt
15. Theorem 2.5.16, p. 216 in Template:Harvtxt.
16. see II.A.8, p. 29 in Template:Harvtxt
17. see Theorem 2.6.23, p. 231 in Template:Harvtxt.
18. see N. Bourbaki, (2004), "Integration I", Springer Verlag, ISBN 3-540-41129-1.
19. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
21. see D. Amir, "On isomorphisms of continuous function spaces". Israel J. Math. 3 (1965), 205–210.
22. M. Cambern, "A generalized Banach-Stone theorem". Proc. Amer. Math. Soc. 17 (1966), 396–400, and "On isomorphisms with small bound". Proc. Amer. Math. Soc. 18 (1967), 1062–1066.
23. H. B. Cohen, "A bound-two isomorphism between C(X) Banach spaces". Proc. Amer. Math. Soc. 50 (1975), 215–217.
24. see for example W. Arveson, (1976), "An Invitation to C*-Algebra", Springer-Verlag, ISBN 0-387-90176-0.
25. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
26. see Template:Harvtxt, p. 25.
27. see E. Bishop and R. Phelps, "A proof that every Banach space is subreflexive". Bull. Amer. Math. Soc. 67 (1961), 97–98.
28. see III.C.14, p. 140 in Template:Harvtxt.
29. see Corollary 2, p. 11 in Template:Harvtxt.
30. see p. 85 in Template:Harvtxt.
31. Rosenthal, Haskell P. (1974), "A characterization of Banach spaces containing ℓ1", Proc. Nat. Acad. Sci. U.S.A. 71:2411–2413. Rosenthal's proof is for real scalars. The complex version of the result is due to L. Dor, in Dor, Leonard E. (1975), "On sequences spanning a complex ℓ1 space", Proc. Amer. Math. Soc. 47:515–516.
32. see p. 201 in Template:Harvtxt.
33. {{#invoke:citation/CS1|citation |CitationClass=citation }}.
34. Odell and Rosenthal, Sublemma p. 378 and Remark p. 379.
35. for more on pointwise compact subsets of the Baire class, see {{#invoke:citation/CS1|citation |CitationClass=citation }}.
36. see Proposition 2.5.14, p. 215 in Template:Harvtxt.
37. see for example p. 49, II.C.3 in Template:Harvtxt.
38. see Corollary 2.8.9, p. 251 in Template:Harvtxt.
39. see Template:Harvtxt p. 3.
40. the question appears p. 238, §3 in Banach's book, Template:Harvtxt.
41. see S. V. Bočkarev, "Existence of a basis in the space of functions analytic in the disc, and some properties of Franklin's system". (Russian) Mat. Sb. (N.S.) 95(137) (1974), 3–18, 159.
42. see P. Enflo, "A counterexample to the approximation property in Banach spaces". Acta Math. 130, 309–317(1973).
43. see R.C. James, "Bases and reflexivity of Banach spaces". Ann. of Math. (2) 52, (1950). 518–527. See also Template:Harvtxt p. 9.
44. see A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires". Mem. Amer. Math. Soc. 1955 (1955), no. 16, 140 pp., and A. Grothendieck, "Résumé de la théorie métrique des produits tensoriels topologiques". Bol. Soc. Mat. São Paulo 8 1953 1–79.
45. see chap. 2, p. 15 in Template:Harvtxt.
46. see chap. 3, p. 45 in Template:Harvtxt.
47. see Example. 2.19, p. 29, and pp. 49–50 in Template:Harvtxt.
48. see Proposition 4.6, p. 74 in Template:Harvtxt.
49. see Pisier, Gilles (1983), "Counterexamples to a conjecture of Grothendieck", Acta Math. 151:181–208.
50. see Szankowski, Andrzej (1981), "B(H) does not have the approximation property", Acta Math. 147: 89–108. Ryan claims that this result is due to Per Enflo, p. 74 in Template:Harvtxt.
51. see Kwapień, S. (1970), "A linear topological characterization of inner-product spaces", Studia Math. 38:277–278.
52. see Lindenstrauss, J. and Tzafriri, L. (1971), "On the complemented subspaces problem", Israel J. Math. 9:263–269.
53. see p. 245 in Template:Harvtxt. The homogeneity property is called "propriété (15)" there. Banach writes: "on ne connaît aucun exemple d'espace à une infinité de dimensions qui, sans être isomorphe avec (L2), possède la propriété (15)".
54. Gowers, W. T. (1996), "A new dichotomy for Banach spaces", Geom. Funct. Anal. 6:1083–1093.
55. see Gowers, W. T. (1994), "A solution to Banach's hyperplane problem", Bull. London Math. Soc. 26:523–530.
56. see Komorowski, Ryszard A. and Tomczak–Jaegermann, Nicole (1995), "Banach spaces without local unconditional structure", Israel J. Math. 89:205–226 and also (1998), "Erratum to: Banach spaces without local unconditional structure", Israel J. Math. 105:85–92.
57. Milyutin, Alekseĭ A. (1966), "Isomorphism of the spaces of continuous functions over compact sets of the cardinality of the continuum". (Russian) Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 2:150–156.
58. Milutin. See also Rosenthal, Haskell P., "The Banach spaces C(K)" in Handbook of the geometry of Banach spaces, Vol. 2, 1547–1602, North-Holland, Amsterdam, 2003.
59. One can take α = ω βn, where β + 1 is the Cantor–Bendixson rank of Template:Mvar, and n > 0 is the finite number of points in the β-th derived set K(β) of Template:Mvar. See Mazurkiewicz, Stefan; Sierpiński, Wacław (1920), "Contribution à la topologie des ensembles dénombrables", Fundamenta Math. 1:17–27.
60. Bessaga, Czesław; Pełczyński, Aleksander (1960), "Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions", Studia Math. 19:53–62.

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