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Uniformly smooth space article
Enflo proved that the class of Banach spaces that admit an equivalent uniformly convex norm coincides with the class of super-reflexive Banach spaces, introduced by Robert C. James. As the class of super-reflexive spaces is self-dual, it follows that the class of Banach spaces that admit an equivalent uniformly convex norm coincides with the class of spaces that admit an equivalent uniformly smooth norm. The Pisier renorming theorem states that a super-reflexive space X admits an equivalent uniformly smooth norm for which the modulus of smoothness &rho;X satisfies, for some constant C and some p &gt; 1


 * $$ \rho_X(t) \le C \, t^p, \quad t > 0.$$

It follows that every super-reflexive space Y admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant c &gt; 0 and some positive real q


 * $$ \delta_Y(\varepsilon) \ge c \, \varepsilon^q, \quad \varepsilon \in (0, 2).$$

If a space admits two equivalent norms, one uniformly convex and one uniformly smooth, the Asplund averaging technique produces another equivalent norm that is both uniformly convex and uniformly smooth.

Super-reflexive space
Informally, a super-reflexive Banach space X is a Banach space such that every Banach space Y, in which all finite dimensional subspaces have a very similar copy within X, must be reflexive. By definition, the space X itself is then reflexive. As an elementary example, every Banach space Y whose two dimensional subspaces are isometric to subspaces of satisfies the parallelogram law, hence Y is a Hilbert space, therefore Y is reflexive. So ℓ2 is super-reflexive.

The formal definition does not use isometries, but almost isometries. A Banach space Y is finitely representable in a Banach space X if for every finite-dimensional subspace Y0 of Y and every &epsilon; &gt; 0, there is a subspace X0 of X such that the multiplicative Banach–Mazur distance between X0 and Y0 satisfies


 * $$d(X_0, Y_0) < 1 + \varepsilon.$$

A Banach space finitely representable in ℓ2 is a Hilbert space. Every Banach space is finitely representable in c0. The space Lp([0, 1]) is finitely representable in ℓp.

A Banach space X is super-reflexive if all Banach spaces Y finitely representable in X are reflexive, or, in other words, if no non-reflexive space Y is finitely representable in X. The notion of ultraproduct of a family of Banach spaces allows for a concise definition: the Banach space X is super-reflexive when its ultrapowers are reflexive.

James proved that a space is super-reflexive if and only if its dual is super-reflexive.

Finite trees in Banach spaces
One of James' characterizations of super-reflexivity uses the growth of separated binary trees. A finite binary tree of height n in a Banach space is a family of 2n&thinsp;+&thinsp;1 &minus; 1 vectors that can be organized in successive generations, starting with generation 0 that consists of a single vector x &empty;, the root node of the tree, followed, for , by the family of 2k vectors forming generation k:


 * $$\{x_{\varepsilon_1, \ldots, \varepsilon_k}\},

\ \ \text{where}\ \ \varepsilon_j = \pm 1, \ j = 1, \ldots, k,$$

that are the child nodes of generation k &minus; 1. Furthermore, each vector that is an internal node of the tree, including the root, is the midpoint between its two child nodes:


 * $$x_{\varepsilon_1, \ldots, \varepsilon_k} = \frac{x_{\varepsilon_1, \ldots, \varepsilon_k, 1} + x_{\varepsilon_1, \ldots, \varepsilon_k, -1}} {2}.$$

Given a positive real number t in (0, 2], the tree is said to be  t-separated if for every internal node, the two child nodes are t-separated in the given space norm:


 * $$ \|x_{\varepsilon_1, \ldots, \varepsilon_k, 1} - x_{\varepsilon_1, \ldots, \varepsilon_k, -1}\| \ge t.$$

The Banach space X is super-reflexive if and only if for every t &isin; (0, 2], there is a number n(t) such that every t-separated tree contained in the unit ball of X has height less than n(t).

Uniformly convex spaces are super-reflexive. Let X be uniformly convex, with modulus of convexity &delta;X. By definition of the modulus of convexity, a t-separated tree of height n, contained in the unit ball, must have all points of generation n &minus; 1 contained in the ball of radius 1 - &delta;X(t) &lt; 1. By induction, it follows that all points of generation n &minus; j are contained in the ball of radius


 * $$ (1 - \delta_X(t))^{j}, \ j = 1, \ldots, n.$$

If the height n was so large that


 * $$ (1 - \delta_X(t))^{n-1} < t / 2, $$

then the two points of the first generation could not be t-separated, contrary to the assumption. This gives the required bound n(t), function of &delta;X(t) only.

Using the tree-characterization, Enflo proved that super-reflexive Banach spaces admit an equivalent uniformly convex norm. Trees in a Banach space are a special instance of vector-valued martingales. Adding techniques from scalar martingale theory, Pisier improved Enflo's result by showing that a super-reflexive space X admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant c &gt; 0 and some real number q &ge; 2,


 * $$ \delta_X(t) \ge c \, t^q, \quad t \in [0, 2].$$

Weak convergences of sequences
A sequence {xn&thinsp;} in a Banach space X is weakly convergent to a vector x &isin; X if f&thinsp;(xn) converges to f&thinsp;(x) for every continuous linear functional f in the dual X&thinsp;′. The sequence {xn&thinsp;} is a weakly Cauchy sequence if f&thinsp;(xn) converges to some scalar limit, for every f in X&thinsp;′. A sequence {fn&thinsp;} in the dual X&thinsp;′ is weakly* convergent to a functional f &isin; X&thinsp;′ if fn&thinsp;(x) converges to f&thinsp;(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 {xn&thinsp;} in X is a weakly Cauchy sequence, the limit L above defines a bounded linear functional on the dual X&thinsp;′, i.e., an element L of the bidual of X. 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. For every measure &mu;, the space L1(&mu;) 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 ℓp, 1 &lt; p &lt; &infin;, 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.

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. This means that ℓ1 satisfies Schur's property. Weakly Cauchy sequences and the ℓ1 basis are the opposite cases of the dichotomy established in the following deep result of H. P. Rosenthal.

Theorem. Let {xn&thinsp;} be a bounded sequence in a Banach space. Either {xn&thinsp;} has a weakly Cauchy subsequence, or it admits a subsequence equivalent to the standard unit vector basis of ℓ1.

Sequences, weak and weak* compactness
When X is separable, the unit ball of the dual is metrizable for the weak* topology, and weak*-compact by Banach–Alaoglu, 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. If the dual X&thinsp;′ 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. A set A in a Banach space is relatively weakly compact if and only if every sequence {an} in A has a weakly convergent subsequence.

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

Hilbert space
A most important example of Banach space is that of a Hilbert space. A Hilbert space H on K = R or C is complete for a norm of the form
 * $$ \|x\| = \sqrt{\langle x, x \rangle}, \ \ \text{where} \ \ (x, y) \in H \times H \rightarrow \langle x, y \rangle$$

is the inner product, a K-valued function on H × H, linear in x and such that
 * $$\langle y, x \rangle = \overline{\langle x, y \rangle}, \ \ \langle x, x \rangle \ge 0, \ \ \text{for all}\ \ x, y \in H.$$

The space L2 is a fundamental example of Hilbert space.

For every vector y in a Hilbert space H,
 * $$x \in H \rightarrow f_y(x) = \langle x, y \rangle$$

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

Hilbert spaces are reflexive. In a Hilbert space H, the weak compactness of the unit ball is extremely often used in the following form: every bounded sequence in H has weakly convergent subsequences.

Banach algebras
A Banach algebra is a Banach space A over or C, together with a structure of algebra over K, such that the product map (a, b) &isin; A &times; A &rarr; a&thinsp;b &isin; A is continuous. An equivalent norm on A can be found so that ǁa&thinsp;bǁ &le; ǁaǁ&thinsp;ǁbǁ for all a, b in A. 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 in the complex plane and continuous on the closure of D. Equipped with the max norm on the closure of D, the disk algebra A(D) is a closed subalgebra of $$C(\overline D)$$. 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 &rarr; a* such that ǁa*aǁ = ǁaǁ2. 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 $$\overline f$$.

Dual space
On a dual space X&thinsp;′, there is a topology weaker than the weak topology of X&thinsp;′, called weak* topology. It is the coarsest topology on X&thinsp;′ for which all evaluation maps x′∈X&thinsp;′ &rarr; x′(x), x&isin;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 of the dual space B&thinsp;′ := {x ∈ X&thinsp;′ | ǁxǁ ≤ 1} 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&thinsp;′ of the dual is a metrizable compact in the weak* topology.

The dual of c0 is isometrically isomorphic to ℓ1. The dual of Lp([0, 1]) is isometrically isomorphic to Lq([0, 1]) when 1 &le; p < ∞ and 1/p + 1/q = 1. 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. The subset P(K) of M(K) consisting of non-negative measures of mass 1 is a 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.

Theorem (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.

The result has been extended by Amir and Cambern 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 equal to 2.

In the commutative Banach algebra C(K), the maximal ideals correspond to kernels of Dirac mesures on K.

Theorem If K is a compact Hausdorff space, then the ideal space Ξ of the Banach algebra C(K) is homeomorphic to K.

Examples of reflexive spaces
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 X, the bidual X&thinsp;′′ is "much larger" than X, namely, the quotient X&thinsp;′′&thinsp;/&thinsp;X is infinite dimensional, and even non separable. However, Robert C. James has constructed an example of a non-reflexive space, usually denoted by J, such that the quotient J&thinsp;′′&thinsp;/&thinsp;J is one dimensional. Furthermore, this space J is isometrically isomorphic to its bidual.

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


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

This function is continuous for the compact topology of K if and only if x&thinsp;′′ is actually in X, considered as subset of X&thinsp;′′. 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&thinsp;′′ is the pointwise limit on K of a sequence {xn&thinsp;} &sub; 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.

Some classification results
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 below the results by Amir and Cambern. Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:

Theorem. Let 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 &alpha; is a countably infinite ordinal. The Banach space C(K) is then isometric to C(&lt;1, &alpha;&gt;). When &alpha;, &beta; are two countably infinite ordinals, and assuming &alpha; &le; &beta;, the spaces C(&lt;1, &alpha;&gt;) and C(&lt;1, &beta;&gt;) are isomorphic if and only if $$\beta < \alpha^\omega.$$ For example, the Banach spaces


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

are mutually non-isomorphic.

Interpolation space article
In order to discuss some of the main results of the theory, it is necessary for the reader to have some familiarity with the theory of Banach spaces.

Suppose that a Banach space X is given as linear subspace of a Hausdorff topological vector space Z. The space X is said to be continuously embedded in Z when the inclusion map from X into Z is continuous. Assume that two Banach spaces X0 and X1 are given, and that they are both continuously embedded in a Hausdorff topological vector space Z. This is called a compatible couple of Banach spaces. One can define norms on X0&thinsp;&cap;&thinsp;X1 and X0&thinsp;+&thinsp;X1 by


 * $$\|u\|_{X_0 \cap X_1} := \max ( \|u\|_{X_0}, \|u\|_{X_1} ),$$


 * $$\|u\|_{X_0 + X_1} := \inf \{ \|u_0\|_{X_0} + \|u_1\|_{X_1} \ : \ u = u_0 + u_1, \; u_0 \in X_0, \; u_1 \in X_1 \}.$$

The following inclusions are all continuous:


 * $$X_0 \cap X_1 \subset X_0, \; X_1 \subset X_0 + X_1.$$

The space Z plays no further role, it was merely a tool that allows to make sense of X0&thinsp;+&thinsp;X1. However the interpolation depends in an essential way from the specific manner in which some elements in X0 and in X1 have been identified as being the same elements in X0&thinsp;&cap;&thinsp;X1. The interest now is to come up with intermediate spaces X between X0 and X1 in the following sense:


 * $$X_0 \cap X_1 \subset X \subset X_0 + X_1,$$

where the two inclusions maps are continuous.


 * Definition. Given two compatible couples (X0, X1) and (Y0, Y1), an interpolation pair is a couple (X, Y) of Banach spaces with the two following properties:
 * The space X is intermediate between X0 and X1, and Y is intermediate between Y0 and Y1.
 * If L is a linear operator from X0&thinsp;+&thinsp;X1 into Y0&thinsp;+&thinsp;Y1, which is continuous from X0 to Y0 and from X1 to Y1, then it is also continuous from X to Y.

The interpolation pair (X, Y) is said to be of exponent &theta; (with 0 < &theta; < 1) if there exists a constant C such that
 * $$\|L\|_{X;Y} \leq C \|L\|_{X_0;Y_0}^{1-\theta} \|L\|_{X_1;Y_1}^{\theta}$$

for all operators L as above. The notation ||L||A, B is for the norm of the operator L as a map from A to B. If C = 1 (which is the smallest possible), one says that (X, Y) is an exact interpolation pair of exponent &theta;.

There are many ways of obtaining interpolation spaces (and the Riesz-Thorin theorem is an example of this for Lp spaces). A method for arbitrary Banach spaces is the complex interpolation method.

Example of Lp
The family of Lp spaces (consisting of complex valued functions) behaves well under complex interpolation. If 1 &le; p0, p1 &le; &infin; and 0 &lt; &theta; &lt; 1, then
 * $$\bigl( L^{p_0}(X, \Sigma, \mu), L^{p_1}(X, \Sigma, \mu) \bigr)_\theta = L^p(X, \Sigma, \mu) \ \ \text{if} \ \ \frac 1 p = \frac{1 - \theta}{p_0} + \frac{1 - \theta}{p_1},$$

with equality of norms, where (X, &Sigma;, &mu;) is an arbitrary measure space.

An important example is that of the couple (L1(X, &Sigma;, &mu;), L&infin;(X, &Sigma;, &mu;)), where the functional K(t, f&thinsp;; L1, L&infin;) can be computed explicitely. The measure &mu; is supposed non-atomic and &sigma;-finite. In this context, the best way of cutting the function f &isin; L1 + L&infin; as sum of two functions f0 in L1 and f1 in L&infin; is to let, for some s > 0 to be chosen as function of t, f1(x) be given for all x &isin; X by
 * $$f_1(x) = f(x) \ \ \text{if} \ \ |f(x)| < s, \ \ \text{and} \ \ f_1(x) = s f(x) / |f(x))| \ \ \text{otherwise}.$$

The optimal choice of s leads to the formula
 * $$K(t, f; L^1, L^\infty) = \int_0^t f^*(u) \, d u,$$

where f&thinsp;* is the decreasing rearrangement of f.

The reiteration theorem
An intermediate space X of the compatible couple (X0, X1) is said to be of class &theta; if
 * $$(X_0, X_1)_{\theta,1} \subset X \subset (X_0, X_1)_{\theta,\infty},\,$$

with continuous injections. Beside all real interpolation spaces (X0, X1)&theta;,&thinsp;q&thinsp; with parameter &theta; and 1 &le; q &le; &infin;, the complex interpolation space (X0, X1)&theta; is an intermediate space of class &theta; of the compatible couple (X0, X1).

The reiteration theorems says, in essence, that interpolating with a parameter &theta; behaves, in some way, like forming a convex combination a = (1&thinsp;-&thinsp;&theta;)&thinsp;x0&thinsp;+&thinsp;&theta;&thinsp;x1&thinsp;: taking a further convex combination of two convex combinations gives another convex combination.

Theorem. Let A0, A1 be intermediate spaces of the compatible couple (X0, X1), of class &theta;0 and &theta;1 respectively, with 0 &lt; &theta;0, &theta;1 &lt; 1 and &theta;0 &ne; &theta;1. When 0 &lt; &theta; &lt; 1 and 1 &le; q &le; &infin;, one has


 * $$(A_0, A_1)_{\theta, q} = (X_0, X_1)_{\eta, q}, \ \ \text{where} \ \ \eta = (1 - \theta) \theta_0 + \theta \, \theta_1.$$

It is notable that when interpolating with the real method between A0 = (X0, X1)&theta; 0,q&thinsp;0 and A1 = (X0, X1)&theta; 1,q&thinsp;1, only the values of &theta;0 and &theta;1 matter. Also, A0 and A1 can be complex interpolation spaces between X0 and X1, with parameters &theta;0 and &theta;1 respectively.

Example
In their introduction, Lions and Peetre say about the reiteration theorem for the real interpolation method: Nous terminons (n&deg; 3) par un analogue &laquo; abstrait &raquo; du théorème de Marcinkiewicz. Indeed, let T be a linear map between Lp spaces that is of weak type (p0, p0) and (p1, p1), with 1 &lt; p0 &lt; p1 &lt; &infin;. This means that T is bounded from


 * $$ L^{p_0} = (L^\infty, L^1)_{\theta_0, p_0} = A_0 \ \ \ \text{to} \ \ \ L^{p_0, \infty} = (L^\infty, L^1)_{\theta_0, \infty} = B_0, \quad \theta_0 = 1 / p_0, $$

and from to Lp1, &infin; = (L&infin;, L1)&theta; 1, &infin; = B1, where &theta;1 = 1 / p1. By the reiteration theorem, for every p &isin; (p0,p1), T is bounded from


 * $$ L^p = (L^\infty, L^1)_{\eta, p} = (A_0, A_1)_{\theta, p}, \quad \eta = \frac 1 p = \frac{1 - \theta}{p_0} + \frac{\theta}{p_1},$$
 * $$ \text{to} \ \ L^p = (L^\infty, L^1)_{\eta, p} = (B_0, B_1)_{\theta, p}.$$

There is also a reiteration theorem for the complex method.

Theorem. Let (X0, X1) be a compatible couple of complex Banach spaces, and assume that X0 &cap; X1 is dense in X0 and in X1. Let and, where 0 &le; &theta;0 &le; &theta;1 &le; 1. Assume further that X0 &cap; X1 is dense in A0 &cap; A1. Then, for every &theta; &isin; [0, 1],


 * $$ \bigl( (X_0, X_1)_{\theta_0}, (X_0, X_1)_{\theta_1} \bigr)_\theta = (X_0, X_1)_\eta, \ \ \text{to} \ \ \eta = (1 - \theta) \theta_0 + \theta \, \theta_1.$$

The density condition is always satisfied when X0 &sub; X1 or X1 &sub; X0.

Duality
Let (X0, X1) be a compatible couple, and assume that X0 &cap; X1 is dense in X0 and in X1. In this case, the restriction map from the (continuous) dual X&thinsp;′j of Xj,, to the dual of X0 &cap; X1 is one-to-one. It follows that the pair of duals (X&thinsp;′0, X&thinsp;′1) is a compatible couple continuously embedded in the dual (X0 &cap; X1)&thinsp;′.

For the complex interpolation method, the following duality result holds:

Theorem. Let (X0, X1) be a compatible couple of complex Banach spaces, and assume that X0 &cap; X1 is dense in X0 and in X1. If X0 and X1 are reflexive, then the dual of the complex interpolation space is obtained by interpolating the duals,


 * $$ ( (X_0, X_1)_\theta )' = (X'_0, X'_1)_\theta, \quad 0 < \theta < 1.$$

In general, the dual of the space (X0, X1)&theta; is equal to (X&thinsp;′0, X&thinsp;′1)&theta;, a space defined by a variant of the complex method. The upper-&theta; and lower-&theta; methods do not coincide in general, but they do for reflexive spaces.

For the real interpolation method, the duality holds provided that the parameter q is finite:

Theorem. Let 0 &lt; &theta; &lt; 1, 1 &le; q &lt; &infin; and (X0, X1) a compatible couple of real Banach spaces. Assume that X0 &cap; X1 is dense in X0 and in X1. Then


 * $$ ( (X_0, X_1)_{\theta, q} )' = (X'_0, X'_1)_{\theta, q'}, \ \ \text{where} \ \ 1/q' = 1 - 1/q.$$

Discrete and abstract definitions
Since the function t &rarr; K(x, t) varies regularly (it is increasing, but K(x, t)&thinsp;/&thinsp;t is decreasing), the previously given definition of the K&theta;,&thinsp;q&thinsp;-norm of a vector x by an integral is equivalent to a definition by a series, obtained by breaking (0, &infin;) into pieces (2n, 2n+1) of equal mass for the measure d t&thinsp;/&thinsp;t,


 * $$ \|x\|_{\theta, q} \simeq \Bigl( \sum_{n \in \mathbf{Z}} \bigl( 2^{-\theta n} K(x, 2^n; X_0, X_1) \bigr)^q \Bigr)^{1/q}.$$

In the important special case where X0 is continuously embedded in X1, one can dispense with the negative part of the sum. In this case, each of the functions x &rarr; K(x, 2n; X0, X1) defines an equivalent norm on X1.

The interpolation space (X0, X1)&theta;,&thinsp;q is a "diagonal subspace" of an ℓq-sum of a sequence of Banach spaces (each one being isomorphic to X0 + X1). Therefore, when q is finite, the dual of (X0, X1)&theta;,&thinsp;q&thinsp; is a quotient of the ℓp-sum of the duals, 1&thinsp;/&thinsp;p + 1&thinsp;/&thinsp;q = 1, which leads to the following form for the discrete J&theta;,&thinsp;q&thinsp;-norm of a functional x' in the dual of (X0, X1)&theta;,&thinsp;q&thinsp;:


 * $$ \inf \Bigl( \sum_{n \in \mathbf{Z}} \bigl( 2^{\theta n} \max(\|x'_n\|_{X'_0}, 2^{-n} \|x'_n\|_{X'_1}) \bigr)^p \Bigr)^{1/p}, \quad 1/p = 1 - 1 / q,$$

where inf is taken over all representations of x' as
 * $$ x' = \sum_{n \in \mathbf{Z}} x'_n.$$

The usual discrete form of the J&theta;,&thinsp;q&thinsp;-norm is obtained by changing n to &minus;n.

The discrete definition makes several questions easier, among which the already mentioned identification of the dual. Other such questions are compactness of operators, in Lions–Peetre, or weak compactness: Davis, Figiel, Johnson and Pełczyński have used interpolation in their proof of the following result:

Theorem. A bounded linear operator between two Banach spaces is weakly compact if and only if it factors through a reflexive space.

Abstract definition
The space ℓq used for the above summation can be replaced by another sequence space Y with unconditional basis, and the weights, , that are used for the K&theta;,&thinsp;q&thinsp;-norm, can be replaced by general weights


 * $$a_n, b_n > 0, \ \ \sum_{n=1}^\infty \min(a_n, b_n) < \infty.$$

The abstract interpolation space K(X0, X1, Y, {an}, {bn}) consists of the vectors x in X0 + X1 such that


 * $$ \|x\|_{K(X_0, X_1)} = \sup_{m \ge 1} \Bigl\| \sum_{n=1}^m a_n K(x, b_n / a_n; X_0, X_1) \, y_n\Bigr\|_Y < \infty,$$

where {yn} is the basis of Y. The abstract method can be used, for example, in order to prove the following result:

Theorem. A Banach space with unconditional basis is isomorphic to a complemented subspace of a space with symmetric basis.

Interpolation of Sobolev and Besov spaces
Several interpolation results are available for Sobolev spaces and Besov spaces on Rn,


 * $$ H^s_p, \ \ s \in \mathbf{R}, \ 1 \le p \le \infty \, ; \quad B^s_{p, q}, \ \ s \in \mathbf{R}, \ 1 \le p, q \le \infty.$$

These spaces are spaces of measurable functions on Rn when s &ge; 0, and of tempered distributions on Rn when s &lt; 0. For the rest of the section, the following setting and notation will be used:


 * $$ 0 < \theta < 1, \ \ 1 \le p, p_0, p_1, q, q_0, q_1 \le \infty, \ \ s, s_0, s_1 \in \mathbf{R},$$

and


 * $$ \frac 1 {p_\theta} = \frac{1 - \theta}{p_0} + \frac{\theta}{p_1}, \ \frac 1 {q_\theta} = \frac{1 - \theta}{q_0} + \frac{\theta}{q_1}, \ \ s_\theta = (1 - \theta) s_0 + \theta s_1.$$

Complex interpolation works well on the class of Sobolev spaces $$H^{s}_{p}$$ (the Bessel potential spaces),


 * $$(H^{s_0}_{p_0}, H^{s_1}_{p_1})_\theta = H^{s_\theta}_{p_\theta}, \quad s_0 \ne s_1, \ 1 < p_0, p_1 < \infty,$$

and it also works well with the class of Besov spaces,


 * $$(B^{s_0}_{p_0, q_0}, B^{s_1}_{p_1, q_1})_\theta = B^{s_\theta}_{p_\theta, q_\theta}, \quad s_0 \ne s_1, \ 1 \le p_0, p_1, q_0, q_1 \le \infty.$$

Real interpolation between Sobolev spaces may give Besov spaces, except when ,


 * $$(H^{s}_{p_0}, H^{s}_{p_1})_{\theta, p_\theta} = H^{s}_{p_\theta}, \quad 1 \le p_0, p_1 \le \infty.$$

When s0 &ne; s1 but, real interpolation between Sobolev spaces gives a Besov space:


 * $$(H^{s_0}_p, H^{s_1}_p)_{\theta, q} = B^{s_\theta}_{p, q}, \quad s_0 \ne s_1, \ 1 \le p, q \le \infty.$$

Also,
 * $$(B^{s_0}_{p, q_0}, B^{s_1}_{p, q_1})_{\theta, q} = B^{s_\theta}_{p, q}, \quad s_0 \ne s_1, \ 1 \le p, q, q_0, q_1 \le \infty,$$

and


 * $$(B^{s}_{p, q_0}, B^{s}_{p, q_1})_{\theta, q} = B^{s}_{p, q_\theta}, \quad 1 \le p, q_0, q_1 \le \infty,$$


 * $$(B^{s_0}_{p_0, q_0}, B^{s_1}_{p_1, q_1})_{\theta, q_\theta} = B^{s_\theta}_{p_\theta, q_\theta}, \quad s_0 \ne s_1, \ p_\theta =q_\theta, \ 1 \le p_0, p_1, q_0, q_1 \le \infty.$$

The Haar functions
For every pair n, k of integers in Z, the Haar function &psi;n,&thinsp;k is defined on the real line R by the formula
 * $$ \psi_{n,k}(t) = 2^{n / 2} \psi(2^n t-k), \quad t \in \mathbf{R}.$$

This function is supported on the right-open interval In, &thinsp;k = [ k&thinsp;2&minus;n, (k+1)&thinsp;2&minus;n&thinsp;), i.e., it vanishes outside that interval. It has integral 0 and norm 1 in the Hilbert space L2(R),
 * $$ \int_{\mathbf{R}} \psi_{n, k}(t) \, d t = 0, \quad \int_{\mathbf{R}} \psi_{n, k}(t)^2 \, d t = 1.$$

The Haar functions are pairwise orthogonal,
 * $$ \int_{\mathbf{R}} \psi_{n_1, k_1}(t) \psi_{n_2, k_2}(t) \, d t = \delta_{n_1, n_2} \delta_{k_1, k_2}, $$

where δi,j represents the Kronecker delta. The reason for orthogonality is that the two supporting intervals $$I_{n_1, k_1}$$ and $$I_{n_2, k_2}$$ are, either disjoint, or else, the smaller of the two supports, say $$I_{n_1, k_1}$$, is contained in the lower or in the upper half of the other, on which the function $$\psi_{n_2, k_2}$$ remains constant.

The Haar system on the real line is the set of functions
 * $$\{ \psi_{n,k}(t) \; ; \; n \in \mathbf{Z}, \; k \in \mathbf{Z} \}.$$

It is complete in L2(R): The Haar system on the line is an orthonormal basis in L2(R).

The Haar system on R
In this section, the space L2([a, b]) will be identified with the subspace of  L2(R) consisting of functions that vanish outside [a, b]. Recall that &psi; is defined on R and vanishes outside [0, 1]. For all integers n, k in Z, let
 * $$ \psi_{n,k}(t) = 2^\frac{n}{2} \psi(2^n t-k), \quad t \in \mathbf{R}.$$

This function is supported on [ k&thinsp;2&minus;n, (k+1)&thinsp;2&minus;n&thinsp;] and has norm 1 in L2(R).

If the function &phi;, indicator function of [0, 1], is removed from the Haar basis of L2([0, 1]), the closed linear span of the remaining functions is the orthogonal of &phi; in L2([0, 1]), namely, the space of functions with integral 0 and supported on [0, 1]. In the same manner, the closed linear span of the set of Haar functions supported on [&minus;1, 0] is the space of functions with integral 0 and supported on [&minus;1, 0]. Putting these two observations together, one sees that the functions &psi;n, k, for n &ge; 0 and &minus;2n &le; k &lt; 2n generate, by means of sums of L2-convergent series, the closed linear subspace of L2(R) defined by
 * $$ F_0 := \{ f \in L^2([-1, 1]) : \int_{-1}^0 f(t) \, d t = \int_0^1 f(t) \, d t = 0 \}.$$

Likewise, for every integer j > 0, the system &psi;n, k, for n + j &ge; 0 and &minus;2n+j &le; k &lt; 2n+j is orthonormal and generates the subspace of L2(R) defined by
 * $$ F_j := \{ f \in L^2([-2^j, 2^j]) : \int_{-2^j}^0 f(t) \, d t = \int_0^{2^j} f(t) \, d t = 0 \}.$$

Since the union of the increasing sequence {Fj} is dense in L2(R), we conclude that the Haar system on the line is an orthonormal basis in L2(R). The density of the union F of the Fj follows from a simple remark: if f is supported on [0, a] and has integral 1 (say), the sequence
 * $$ f_n := f - n^{-1} \mathbf{1}_{[0, n]}$$

consists of functions of F and tends to f in L2.

The Schauder system and the Franklin system
The Schauder system is the family of continuous functions on [0, 1] consisting of the constant function 1, and of multiples of indefinite integrals of the functions in the Haar system on [0, 1], chosen to have norm 1 in the maximum norm. This system begins with s0 = 1, then is the indefinite integral vanishing at 0 of the function 1, first element of the Haar system on [0, 1]. Next, for every integer n &ge; 0, functions sn,&thinsp;k are defined by the formula

s_{n, k}(t) = 2^{1 + n/2} \int_0^t \psi_{n, k}(u) \, d u, \quad t \in [0, 1], \ 0 \le k < 2^n.$$ These functions sn,&thinsp;k are continuous, piecewise linear, supported by the interval In,&thinsp;k that also supports &psi;n,&thinsp;k. The function sn,&thinsp;k is equal to 1 at the midpoint xn,&thinsp;k of the interval In,&thinsp;k, linear on both halves of that interval. It takes values between 0 and 1 everywhere.

The Schauder system is a Schauder basis for the space C([0, 1]) of continuous functions on [0, 1]. For every f in C([0, 1]), the partial sum
 * $$ f_{n+1} = a_0 s_0 + a_1 s_1 + \sum_{m = 0}^{n-1} \Bigl( \sum_{k=0}^{2^m - 1} a_{m,k} s_{m, k} \Bigr) \in C([0, 1])$$

of the series expansion of f in the Schauder system is the continuous piecewise linear function that agrees with f at the 2n&thinsp;+&thinsp;1 points k&thinsp;2&minus;n, where 0 &le; k &le; 2n. Next, the formula
 * $$ f_{n+2} - f_{n+1} = \sum_{k=0}^{2^n - 1} \bigl( f(x_{n,k}) - f_{n+1}(x_{n, k}) \bigr) s_{n, k} = \sum_{k=0}^{2^n - 1} a_{n, k} s_{n, k} $$

gives a way to compute the expansion of f step by step. Since f is uniformly continuous, the sequence {fn} converges uniformly to f, i.e., the sum of the Schauder series expansion of f is equal to f.

The Franklin system is obtained from the Schauder system by the Gram–Schmidt orthonormalization procedure. Since the Franklin system has the same linear span as that of the Schauder system, this span is dense in C([0, 1]), hence in L2([0, 1]). The Franklin system is therefore an orthonormal basis for L2([0, 1]), consisting of continuous piecewise linear functions. P. Franklin proved in 1928 that this system is a Schauder basis for C([0, 1]). The Franklin system is also an unconditional basis for the space Lp([0, 1]) when 1 &lt; p &lt; ∞. It is a Schauder basis in the disk algebra A(D): this was proved in 1974 by Bočkarev, after the existence of a basis for the disk algebra had remained an open question for several years.

Bočkarev's construction of the basis in A(D) goes as follows: let f be a Lipschitz function on [0, &pi;]; then f is the sum of a cosine series with absolutely summable coefficients. Let T(f) be the Taylor series with the same coefficients,


 * $$ \bigl\{ f : x \in [0, \pi] \rightarrow \sum_{n=0}^\infty a_n \cos(n x) \bigr\} \ \longrightarrow \ \bigl\{ T(f) : z \rightarrow \sum_{n=0}^\infty a_n z^n, \quad |z| \le 1 \bigr\}.$$

The basis of A(D) is formed by the images under T of the functions in the Franklin system on [0, &pi;]. Bočkarev's equivalent description for the mapping T starts by extending f to an even Lipschitz function g1 on [&minus;&pi;, &pi;], identified with a Lipschitz function on the unit circle T. Next, let g2 be the conjugate function of g1, and define T(f) to be the function in A(D) whose value on the boundary T of D is equal to g1 + i&thinsp;g2.

Schauder basis article
A Schauder basis in a Banach space $X$ is a sequence &#123;$e_{n}$&#125;$_{n&ge;0}$ of vectors in $X$ with the property that for every vector $x$ in $X$, there exist uniquely defined scalars &#123;$a_{n}$&#125;$_{n&ge;0}$ depending on $x$, such that
 * $$ x = \sum_{n=0}^{+\infty} a_n e_n, \ \ \textit{i.e.,} \ \ x = \lim_n P_n(x), \ \ P_n(x) := \sum_{k=0}^n a_k e_k.$$

It follows from the Banach–Steinhaus theorem that the linear mappings &#123;$P_{n}$&#125; are uniformly bounded by some constant $C$. When the basis vectors have norm 1, the coordinate functionals $e*_{n}$ which assign to every $x$ in $X$ the coordinate $a_{n}$ of $x$ in the above expansion have norm $&le; 2C$ in the dual of $X$. They are called biorthogonal functionals.

Most classical spaces have explicit bases. The Haar system &#123;$h_{n}$&#125; is a basis for Lp([0, 1]), 1$&le; p < ∞$. The trigonometric system is a basis in C(T) when 1$< p < ∞$. Let the Schauder system be the family of continuous functions on [0, 1] consisting of the function 1 and of all functions $f_{n}$ such that $f_{n}$(0) = 0 and that the derivative of $f_{n}$ is equal to $h_{n}$. In the space C([0, 1]), the Schauder system is a basis.

Since every $x$ in $X$ is the limit of $P_{n}$($x$), with $P_{n}$ of finite rank and uniformly bounded, the space $X$ satisfies the uniform approximation property. The first example by Enflo of a space failing the approximation property was at the same time the first example of a Banach space without basis.

A basis is boundedly complete if the sequence {Pn($x$)} converges in $X$ whenever it is bounded. The unit vector basis for ℓp, 1$&le; p < ∞$, is boundedly complete, while it is not boundedly complete in c0.

A basis is shrinking if for every bounded linear functional $f$ on $X$, the sequence
 * $$\sup \{|f(x)| : x \in F_n, \; \|x\| \le 1 \}$$

tends to 0, where $F_{n}$ is the linear span of the basis vectors $e_{m}$ for $m &ge; n$. The unit vector basis for ℓp, 1$< p < ∞$, or for c0, is shrinking. It is not shrinking in ℓ1.

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

Unconditional bases

ℓ∞

Bases for spaces of operators
The space K(ℓ2) of compact operators on the Hilbert space ℓ2 has a Schauder basis. For every x, y in ℓ2, let x ⊗ y denote the rank one operator v &isin; ℓ2 &rarr; &lt;v,&thinsp;x&gt;&thinsp;y. If {en&thinsp;}n&thinsp;&ge;&thinsp;1 is the standard orthonormal basis of ℓ2, a basis for K(ℓ2) is given by the sequence


 * $$\begin{align} & e_1 \otimes e_1, \ \ e_1 \otimes e_2, \; e_2 \otimes e_2, \; e_2 \otimes e_1, \ldots, \\

& e_1 \otimes e_n, e_2 \otimes e_n, \ldots, e_n \otimes e_n, e_n \otimes e_{n-1}, \ldots, e_n \otimes e_1, \ldots \end{align}$$

For every n, the sequence consisting of the n2 first vectors in this basis is a special ordering of the family {ej ⊗ ek}, for 1 &le; j, k &le; n.

The preceding result can be generalized: a Banach space X with a basis has the approximation property, so the space K(X) of compact operators on X is isometrically isomorphic to the injective tensor product
 * $$X' \widehat \otimes_\varepsilon X \simeq \mathcal{K}(X).$$

If X is a Banach space with a Schauder basis {en&thinsp;}n&thinsp;&ge;&thinsp;1 such that the biorthogonal functionals are a basis of the dual, that is to say, a Banach space with a shrinking basis, then the space K(X) admits a basis formed by the rank one operators e&thinsp;*j ⊗ ek : v &rarr; e&thinsp;*j&thinsp;(v)&thinsp;ek, with the same ordering as before. This applies in particular to every reflexive Banach space X with a Schauder basis

On the other hand, the space B(ℓ2) has no basis, since it is non-separable. Moreover, B(ℓ2) does not have the approximation property.

Convolution article
The Young inequality for convolutions is also true in other contexts (circle group, convolution on Z). The preceding inequality is not sharp on the real line: when 1 &lt; p, q, r &lt; &infin;, there exists a constant Cp, q &lt; 1 such that the Lr(R) norm of f ∗ g is bounded by Cp, q times the product of norms ||f||p ||g||q. The optimal value of Cp, q was discovered in 1975.