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] 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 X 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 {x_n} in X, there exists an element x in X such that

$\lim_{n\to\infty}x_n=x,$

or equivalently:

$\lim_{n\to\infty}\left \|x_n - x \right \|_X = 0.$

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

$\sum_{n=1}^{\infty} \|v_n\|_X < \infty \quad \text{implies that} \quad \sum_{n=1}^{\infty} v_n\ \ \text{converges in} \ \ X.$

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

If X and Y are normed spaces over the same ground field K, the set of all continuous K-linear maps T : X → Y 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

$\|T\| = \sup \left \{\|Tx\|_Y \mid x\in X,\ \|x\|_X\le 1 \right \}.$

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 B(X) = B(X, X) 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 T : X → Y such that T and its inverse T⁻¹ 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, i.e., ||T(x)|| = ||x|| for every x in X. The Banach-Mazur distance d(X, Y) between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and Y differ.

Basic notions

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

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

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

$\|(x, y)\|_1 = \|x\| + \|y\|, \qquad \|(x, y)\|_\infty = \max (\|x\|, \|y\|)$

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

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

$\| x + M\| = \inf\limits_{m \in M} \|x+m\|.$

The quotient X / M is a Banach space when X is complete.^[5] The quotient map from X onto X / M, sending x in X 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 M of X is said to be a complemented subspace of X if M is the range of a bounded linear projection P from X onto M. In this case, the space X is isomorphic to the direct sum of M and Ker(P), the kernel of the projection P.

Suppose that X and Y are Banach spaces and that T ∈ B(X, Y). There exists a canonical factorization of T as^[5]

$T = T_1 \circ \pi, \ \ \ T : X \ \overset{\pi}{\longrightarrow}\ X / \operatorname{Ker}(T) \ \overset{T_1}{\longrightarrow} \ Y$

where the first map π is the quotient map, and the second map T₁ sends every class x + Ker(T) in the quotient to the image T(x) in Y. 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 L^p spaces and their special cases, the sequence spaces ℓ^p that consist of scalar sequences indexed by N; among them, the space ℓ¹ of absolutely summable sequences and the space ℓ² of square summable sequences; the space c₀ of sequences tending to zero and the space ℓ^∞ of bounded sequences; the space C(K) of continuous scalar functions on a compact Hausdorff space K, equipped with the max norm,

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

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 X, there is a closed subspace M of ℓ¹ such that X ≅ ℓ¹/M.^[8]

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

$\|x\|_H = \sqrt{\langle x, x \rangle},$

where

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

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

$\begin{align} \forall x, y \in H: \quad \langle y, x \rangle &= \overline{\langle x, y \rangle}, \\ \forall x \in H: \quad \langle x, x \rangle &\ge 0, \\ \langle x,x \rangle = 0 \Leftrightarrow x &= 0. \end{align}$

For example, the space L² is a Hilbert space.

The Hardy spaces, the Sobolev spaces are examples of Banach spaces that are related to L^p 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 A over K = R or C, together with a structure of algebra over K, such that the product map (a, b) ∈ A × A → ab ∈ A is continuous. An equivalent norm on A can be found so that ||ab|| ≤ ||a|| ||b|| for all a, b ∈ A.

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 D ⊂ C 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 ℓ¹(Z), 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 a → a^∗ such that ||a^∗a|| = ||a||². 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 C(K) of complex continuous functions on a compact Hausdorff space K is an example of commutative C*-algebra, where the involution associates to every function  f  its complex conjugate  f .

Dual space

If X 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 X 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 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 K = R, C. Let further

• Y ⊆ X be a linear subspace,
• p : X → R be a sublinear function and
•  f  : Y → K be a linear functional so that Re( f (y)) ≤ p(y) for all y in Y.

Then, there exists a linear functional F : X → K so that

$F|_Y=f, \quad \text{and} \quad \forall x\in X, \ \ \operatorname{Re}(F(x))\leq p(x).$

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 x in a normed space X, there exists a continuous linear functional  f  on X such that

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

When x is not equal to the 0 vector, the functional  f  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.^[11]

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.^[12] If M is a closed linear subspace in X, one can associate the orthogonal of M in the dual,

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

The orthogonal M^⊥ is a closed linear subspace of the dual. The dual of M 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 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 topologies

The weak topology on a Banach space X is the coarsest topology on X 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 X and Y is also weakly continuous, i.e., continuous from the weak topology of X to that of Y.^[16]

If X is infinite-dimensional, there exist linear maps which are not continuous. The space X^∗ of all linear maps from X to the underlying field K (this space X^∗ is called the algebraic dual space, to distinguish it from X ′) also induces a topology on X 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 X be a normed vector space. Then the closed unit ball B ′ = {x′ ∈ X ′ : ||x′|| ≤ 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 X 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 ℓ¹: for every bounded linear functional  f  on c₀, there is a unique element y = {y_n} ∈ ℓ¹ such that

$f(x) = \sum_{n \in \mathbf{N}} x_n y_n, \qquad x = \{x_n\} \in c_0, \ \ \text{and} \ \ \|f\|_{(c_0)'} = \|y\|_{\ell_1}.$

The dual of ℓ¹ is isometrically isomorphic to ℓ^∞. The dual of L^p([0, 1]) is isometrically isomorphic to L^q([0, 1]) when 1 ≤ p < ∞ and  1/p + 1/q = 1.

For every vector y in a Hilbert space H, the mapping

$x \in H \to f_y(x) = \langle x, y \rangle$

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

When K is a compact Hausdorff topological space, the dual M(K) of C(K) is the space of Radon measures in the sense of Bourbaki.^[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 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 C(K) and C(L) are isometrically isomorphic, then the topological spaces K and L 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 K,

$I_x = \ker \delta_x = \{f \in C(K) : f(x) = 0\}, \quad x \in K.$

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

Not every unital commutative Banach algebra is of the form C(K) for some compact Hausdorff space K. However, this statement holds if one places C(K) in the smaller category of commutative C*-algebras. Gelfand's representation theorem for commutative C*-algebras states that every commutative unital C*-algebra A is isometrically isomorphic to a C(K) space.^[24] The Hausdorff compact space K here is again the maximal ideal space, also called the spectrum of A in the C*-algebra context.

Bidual

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

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

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

For example, the dual of X = c₀ is identified with ℓ¹, and the dual of ℓ¹ is identified with ℓ^∞, the space of bounded scalar sequences. Under these identifications, F_X is the inclusion map from c₀ to ℓ^∞. It is indeed isometric, but not onto.

If F_X is surjective, then the normed space X 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 F_X, 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 X ′′. If X is not reflexive, the unit ball of X 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 {x_j} in X so that

$\sup_j \|x_j\| \le \|x''\|, \ \ x''(f) = \lim_j f(x_j), \quad f \in X'.$

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 ℓ¹ that are not in ℓ¹ cannot be weak*-limit of sequences in ℓ¹, since ℓ¹ 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 (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 sup_T∈F ||T(x)||_Y < ∞, then sup_T∈F ||T||_Y < ∞.

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,

$\sup_{T \in F} \sup_{x \in U} \; \|T(x)\|_Y < \infty.$

The Open Mapping Theorem. Let X and Y be Banach spaces and T : X → Y be a continuous linear operator. Then T is surjective if and only if 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 T ∈ B(X, Y). Suppose further that the range of T is closed in Y. 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 X is the internal direct sum of closed subspaces M₁, ..., M_n, then X is isomorphic to M₁ ⊕ ... ⊕ M_n.

This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from M₁ ⊕ ... ⊕ M_n onto X sending (m₁, ..., m_n) to the sum m₁ + ... + m_n.

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

Reflexivity

The normed space X is called reflexive when the natural map

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

is surjective. Reflexive normed spaces are Banach spaces.

Theorem. If 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 X₁, ..., X_n are normed spaces and that X = X₁ ⊕ ... ⊕ X_n. Then X is reflexive if and only if each X_j is reflexive.

Hilbert spaces are reflexive. The L^p 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 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^[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 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 R, every continuous linear functional  f  in X ′ attains its maximum || f || 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  f  in X ′ there exists x in X with ||x|| ≤ 1, so that  f (x) = || f ||.

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 Bishop–Phelps theorem^[27] states that norm-attaining functionals are norm dense in the dual X ′ of X.

Weak convergences of sequences

A sequence {x_n} in a Banach space X is weakly convergent to a vector x ∈ X if  f (x_n) converges to  f (x) for every continuous linear functional  f  in the dual X ′. The sequence {x_n} is a weakly Cauchy sequence if  f (x_n) converges to a scalar limit L( f ), for every  f  in X ′. A sequence { f_n } in the dual X ′ is weakly* convergent to a functional  f  ∈ X ′ if  f_n (x) converges to  f (x) 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 {x_n} in X is a weakly Cauchy sequence, the limit L above defines a bounded linear functional on the dual X ′, i.e., an element L of the bidual of X, and L is the limit of {x_n} 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. ^[28] For every measure μ, 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 ℓ^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 ℓ¹ is not weakly Cauchy. Weakly Cauchy sequences in ℓ¹ are weakly convergent, since L¹-spaces are weakly sequentially complete. Actually, weakly convergent sequences in ℓ¹ are norm convergent.^[30] This means that ℓ¹ satisfies Schur's property.

Results involving the ℓ¹ basis

Weakly Cauchy sequences and the ℓ¹ basis are the opposite cases of the dichotomy established in the following deep result of H. P. Rosenthal.^[31]

Theorem.^[32] Let {x_n} be a bounded sequence in a Banach space. Either {x_n} has a weakly Cauchy subsequence, or it admits a subsequence equivalent to the standard unit vector basis of ℓ¹.

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

Theorem.^[33] Let X be a separable Banach space. The following are equivalent:

• The space X does not contain a closed subspace isomorphic to ℓ¹.
• Every element of the bidual X ′′ is the weak*-limit of a sequence {x_n} 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 ℓ¹, every element of B ′′ is weak*-limit of a sequence in the unit ball of X.^[34]

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,^[17] and every element x ′′ in the bidual X ′′ defines a bounded function on K:

$x' \in K \mapsto x''(x'), \quad \left |x''(x') \right | \le \left \|x'' \right \|.$

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 ℓ¹. By the preceding result of Odell and Rosenthal, the function x ′′ is the pointwise limit on K of a sequence {x_n} ⊂ X 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.^[35]

Sequences, weak and weak* compactness

When X 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 X is metrizable if and only if X is finite-dimensional.^[36] 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.^[37] A set A in a Banach space is relatively weakly compact if and only if every sequence {a_n} 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.^[38]

A weakly compact subset A in ℓ¹ is norm-compact. Indeed, every sequence in A has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of ℓ¹.

Schauder bases

A Schauder basis in a Banach space X is a sequence {e_n}_n ≥ 0 of vectors in X with the property that for every vector x in X, there exist uniquely defined scalars {x_n}_n ≥ 0 depending on x, such that

$x = \sum_{n=0}^{\infty} x_n e_n, \quad \textit{i.e.,} \quad x = \lim_n P_n(x), \ P_n(x) := \sum_{k=0}^n x_k e_k.$

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 {P_n} are uniformly bounded by some constant C. Let {e∗
n} denote the coordinate functionals which assign to every x in X the coordinate x_n 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 ≤ 2C in the dual of X.

Most classical separable spaces have explicit bases. The Haar system {h_n} is a basis for L^p([0, 1]), 1 ≤ p < ∞. The trigonometric system is a basis in L^p(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 x in a Banach space X with a basis is the limit of P_n(x), with P_n of finite rank and uniformly bounded, the space X 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 separable Banach space without a Schauder basis.

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.^[43] In this case, the biorthogonal functionals form a basis of the dual of X.

Tensor product

Let X and Y be two K-vector spaces. The tensor product X ⊗ Y of X and Y is a K-vector space Z with a bilinear mapping T : X × Y → Z which has the following universal property:

If T₁ : X × Y → Z₁ is any bilinear mapping into a K-vector space Z₁, then there exists a unique linear mapping  f  : Z → Z₁ such that T₁ = f ∘ T.

The image under T of a couple (x, y) in X × Y is denoted by x ⊗ y, and called a simple tensor. Every element z in X ⊗ Y 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 say that the projective tensor product^[45] of two Banach spaces X and Y is the completion $X \widehat{\otimes}_\pi Y$ of the algebraic tensor product X ⊗ Y equipped with the projective tensor norm, and similarly for the injective tensor product^[46] $X \widehat{\otimes}_\varepsilon Y$. Grothendieck proved in particular that^[47]

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

where K is a compact Hausdorff space, C(K, Y) the Banach space of continuous functions from K to Y and L¹([0, 1], Y) the space of Bochner-measurable and integrable functions from [0, 1] to Y, 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 s ∈ K →  f (s)y ∈ Y.

Tensor products and the approximation property

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

For every Banach space Y, there is a natural norm 1 linear map

$Y \widehat\otimes_\pi X \to Y \widehat\otimes_\varepsilon X$

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 Y is the dual of X. Precisely, for every Banach space X, the map

$X' \widehat \otimes_\pi X \ \longrightarrow X' \widehat \otimes_\varepsilon X$

is one-to-one if and only if X has the approximation property.^[48]

Grothendieck conjectured that $X \widehat{\otimes}_\pi Y$ and $X \widehat{\otimes}_\varepsilon Y$ must be different whenever X and Y are infinite-dimensional Banach spaces. This was disproved by Gilles Pisier in 1983.^[49] Pisier constructed an infinite-dimensional Banach space X such that $X \widehat{\otimes}_\pi X$ and $X \widehat{\otimes}_\varepsilon X$ are equal. Furthermore, just as Enflo's example, this space X 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 X to be associated to an inner product is the parallelogram identity:

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

It follows, for example, that the Lebesgue space L^p([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:

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

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

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

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 x. It follows that it is linear over the rationals, thus linear by continuity.

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

$c^{-2} \sum_{k=1}^n \left \|x_k \right \|^2 \le \operatorname{Ave}_{\pm} \left \| \sum_{k=1}^n \pm x_k \right \|^2 \le c^2 \sum_{k=1}^n \left \|x_k \right \|^2$

for every integer n and all families of vectors {x₁, ..., x_n} ⊂ X, then the Banach space X 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 n, any finite-dimensional normed space, with dimension sufficiently large compared to n, contains subspaces nearly isometric to the n-dimensional Euclidean space.

The next result gives the solution of the so-called homogeneous space problem. An infinite-dimensional Banach space X is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces. A Banach space isomorphic to ℓ² 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 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.^[55] 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,^[56] that X is isomorphic to ℓ².

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 K 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 K is homeomorphic to some closed interval of ordinal numbers

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

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 α ≤ β, the spaces C(<1, α >) and C(<1, β >) are isomorphic if and only if β < α^ω.^[60] For example, the Banach spaces

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

are mutually non-isomorphic.

Examples

A glossary of symbols:

• K = R, C;
• X is a compact Hausdorff space;
• I is a closed and bounded interval [a, b];
• p, q are real numbers with 1 < p, q < ∞ so that  1/p + 1/q = 1.
• Σ is a σ-algebra of sets;
• Ξ is an algebra of sets (for spaces only requiring finite additivity, such as the ba space);
• μ is a measure with variation |μ|.

Classical Banach spaces
	Dual space	Reflexive	weakly sequentially complete	Norm	Notes
K^n	K^n	Yes	Yes	$\|x\|_2 = \left(\sum_{i=1}^n |x_i|^2\right)^{\frac{1}{2}}$	Euclidean space
ℓn p	ℓn q	Yes	Yes	$\|x\|_p = \left(\sum_{i=1}^n |x_i|^p\right)^{\frac{1}{p}}$
ℓn ∞	ℓn 1	Yes	Yes	$\|x\|_\infty = \max\nolimits_{1\le i\le n} |x_i|$
ℓp	ℓq	Yes	Yes	$\|x\|_p = \left(\sum_{i=1}^\infty |x_i|^p\right)^{\frac{1}{p}}$
ℓ1	ℓ∞	No	Yes	$\|x\|_1 = \sum_{i=1}^\infty |x_i|$
ℓ∞	ba	No	No	$\|x\|_\infty = \sup\nolimits_i |x_i|$
c	ℓ1	No	No	$\|x\|_\infty = \sup\nolimits_i |x_i|$
c0	ℓ1	No	No	$\|x\|_\infty = \sup\nolimits_i |x_i|$	Isomorphic but not isometric to c.
bv	ℓ∞	No	Yes	$\|x\|_{bv} = |x_1| + \sum_{i=1}^\infty|x_{i+1}-x_i|$	Isometrically isomorphic to ℓ1.
bv0	ℓ∞	No	Yes	$\|x\|_{bv_0} = \sum_{i=1}^\infty|x_{i+1}-x_i|$	Isometrically isomorphic to ℓ1.
bs	ba	No	No	$\|x\|_{bs} = \sup\nolimits_n\left|\sum_{i=1}^nx_i\right|$	Isometrically isomorphic to ℓ∞.
cs	ℓ1	No	No	$\|x\|_{bs} = \sup\nolimits_n\left|\sum_{i=1}^nx_i\right|$	Isometrically isomorphic to c.
B(X, Ξ)	ba(Ξ)	No	No	$\|f\|_B = \sup\nolimits_{x\in X}|f(x)|$
C(X)	rca(X)	No	No	$\|x\|_{C(X)} = \max\nolimits_{x\in X} |f(x)|$
ba(Ξ)	?	No	Yes	$\|\mu\|_{ba} = \sup\nolimits_{A\in\Sigma} |\mu|(A)$
ca(Σ)	?	No	Yes	$\|\mu\|_{ba} = \sup\nolimits_{A\in\Sigma} |\mu|(A)$	A closed subspace of ba(Σ).
rca(Σ)	?	No	Yes	$\|\mu\|_{ba} = \sup\nolimits_{A\in\Sigma} |\mu|(A)$	A closed subspace of ca(Σ).
L^p(μ)	L^q(μ)	Yes	Yes	$\|f\|_p = \left (\int |f|^p\,d\mu\right)^{\frac{1}{p}}$
L^1(μ)	L^∞(μ)	No	Yes	$\|f\|_1 = \int |f|\,d\mu$	The dual is L^∞(μ) if μ is σ-finite.
BV(I)	?	No	Yes	$\|f\|_{BV} = V_f(I) + \lim\nolimits_{x\to a^+}f(x)$	Vf (I) is the total variation of  f
NBV(I)	?	No	Yes	$\|f\|_{BV} = V_f(I)$	NBV(I) consists of BV(I) functions such that $\lim\nolimits_{x\to a^+}f(x)=0$
AC(I)	K + L^∞(I)	No	Yes	$\|f\|_{BV} = V_f(I) + \lim\nolimits_{x\to a^+}f(x)$	Isomorphic to the Sobolev space W^1,1(I).
C^n([a, b])	rca([a,b])	No	No	$\|f\| = \sum_{i=0}^n \sup\nolimits_{x\in [a,b]} \left |f^{(i)}(x) \right |$	Isomorphic to R^n ⊕ 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 R → R, 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.

See also

• Space (mathematics)
  • Hilbert space
  • Lp space
  • Sobolev space
  • Hardy space
• Interpolation space
• Distortion problem

Notes

1. Bourbaki 1987, V.86
2. see Theorem 1.3.9, p. 20 in Megginson (1998).
3. see Corollary 1.4.18, p. 32 in Megginson (1998).
4. see Banach (1932), p. 182.
5. see pp. 17–19 in Carothers (2005).
6. see Banach (1932), pp. 11-12.
7. see Banach (1932), Th. 9 p. 185.
8. see Theorem 6.1, p. 55 in Carothers (2005)
9. Several books about functional analysis use the notation X^∗ for the continuous dual, for example Carothers (2005), Lindenstrauss & Tzafriri (1977), Megginson (1998), Ryan (2002), Wojtaszczyk (1991).
10. Theorem 1.9.6, p. 75 in Megginson (1998)
11. see also Theorem 2.2.26, p. 179 in Megginson (1998)
12. see p. 19 in Carothers (2005).
13. Theorems 1.10.16, 1.10.17 pp.94–95 in Megginson (1998)
14. Theorem 1.12.11, p. 112 in Megginson (1998)
15. Theorem 2.5.16, p. 216 in Megginson (1998).
16. see II.A.8, p. 29 in Wojtaszczyk (1991)
17. see Theorem 2.6.23, p. 231 in Megginson (1998).
18. see N. Bourbaki, (2004), "Integration I", Springer Verlag, ISBN 3-540-41129-1.
19. Eilenberg, Samuel (Jan 22, 1942). "Banach Space Methods in Topology". Annals of Mathematics 43 (3): 568. doi:10.2307/1968812.
20. see also Banach (1932), p. 170 for metrizable K and L.
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. R. C. James (1951). "A non-reflexive Banach space isometric with its second conjugate space". Proc. Natl. Acad. Sci. U.S.A. 37: 174–177. doi:10.1073/pnas.37.3.174.
26. see Lindenstrauss & Tzafriri (1977), 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 Wojtaszczyk (1991).
29. see Corollary 2, p. 11 in Diestel (1984).
30. see p. 85 in Diestel (1984).
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 Diestel (1984).
33. Odell, Edward W.; Rosenthal, Haskell P. (1975), "A double-dual characterization of separable Banach spaces containing ℓ¹", Israel J. Math. 20: 375–384, doi:10.1007/bf02760341.
34. Odell and Rosenthal, Sublemma p. 378 and Remark p. 379.
35. for more on pointwise compact subsets of the Baire class, see Bourgain, Jean; Fremlin, D. H.; Talagrand, Michel (1978), "Pointwise Compact Sets of Baire-Measurable Functions", Am. J. Math. 100: 845–886, doi:10.2307/2373913, JSTOR 2373913.
36. see Proposition 2.5.14, p. 215 in Megginson (1998).
37. see for example p. 49, II.C.3 in Wojtaszczyk (1991).
38. see Corollary 2.8.9, p. 251 in Megginson (1998).
39. see Lindenstrauss & Tzafriri (1977) p. 3.
40. the question appears p. 238, §3 in Banach's book, Banach (1932).
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 Lindenstrauss & Tzafriri (1977) 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 Ryan (2002).
46. see chap. 3, p. 45 in Ryan (2002).
47. see Example. 2.19, p. 29, and pp. 49–50 in Ryan (2002).
48. see Proposition 4.6, p. 74 in Ryan (2002).
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 Ryan (2002).
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 Banach (1932). 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 K, and n > 0 is the finite number of points in the β-th derived set K^(β) of K. See Mazurkiewicz, Stefan; Sierpiński, Wacław (1920), "Contribution à la topologie des ensembles dénombrables", Fundamenta Mathematicae 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.

References

• Banach, Stefan (1932), Théorie des opérations linéaires, Monografie Matematyczne 1, Warszawa: Subwencji Funduszu Kultury Narodowej, Zbl 0005.20901.
• Beauzamy, Bernard (1985) [1982], Introduction to Banach Spaces and their Geometry (Second revised ed.), North-Holland.
• Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-13627-9.
• Carothers, Neal L. (2005), A short course on Banach space theory, London Mathematical Society Student Texts 64, Cambridge: Cambridge University Press, pp. xii+184, ISBN 0-521-84283-2.
• Diestel, Joseph (1984), Sequences and series in Banach spaces, Graduate Texts in Mathematics 92, New York: Springer-Verlag, pp. xii+261, ISBN 0-387-90859-5.
• Dunford, Nelson; Schwartz, Jacob T. with the assistance of W. G. Bade and R. G. Bartle (1958), Linear Operators. I. General Theory, Pure and Applied Mathematics 7, New York: Interscience Publishers, Inc., MR 0117523
• Lindenstrauss, Joram; Tzafriri, Lior (1977), Classical Banach Spaces I, Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Berlin: Springer-Verlag, ISBN 3-540-08072-4.
• Megginson, Robert E. (1998), An introduction to Banach space theory, Graduate Texts in Mathematics 183, New York: Springer-Verlag, pp. xx+596, ISBN 0-387-98431-3.
• Ryan, Raymond A. (2002), Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, London: Springer-Verlag, pp. xiv+225, ISBN 1-85233-437-1.
• Wojtaszczyk, Przemysław (1991), Banach spaces for analysts, Cambridge Studies in Advanced Mathematics 25, Cambridge: Cambridge University Press, pp. xiv+382, ISBN 0-521-35618-0.

External links

• Hazewinkel, Michiel, ed. (2001), "Banach space", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Weisstein, Eric W., "Banach Space", MathWorld.