Interpolation space
In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other Banach spaces. The main applications are in Sobolev spaces, where spaces of functions that have a noninteger number of derivatives are interpolated from the spaces of functions with integer number of derivatives.

History
The theory of interpolation of vector spaces began by an observation of Józef Marcinkiewicz, later generalized and now known as the Riesz-Thorin theorem. In simple terms, if a linear function is continuous on a certain space  and also on a certain space , then it is also continuous on the space , for any intermediate  between  and . In other words,  is a space which is intermediate between  and .

In the development of Sobolev spaces, it became clear that the trace spaces were not any of the usual function spaces (with integer number of derivatives), and Jacques-Louis Lions discovered that indeed these trace spaces were constituted of functions that have a noninteger degree of differentiability.

Many methods were designed to generate such spaces of functions, including the Fourier transform, complex interpolation, 
real interpolation, 
as well as other tools (see e.g. fractional derivative).

The setting of interpolation 
A Banach space  is said to be continuously embedded in a Hausdorff topological vector space  when  is a linear subspace of  such that the inclusion map from  into  is continuous. A compatible couple  of Banach spaces consists of two Banach spaces  and  that are continuously embedded in the same Hausdorff topological vector space . The embedding in a linear space  allows to consider the two linear subspaces

and

Interpolation does not depend only upon the isomorphic (nor isometric) equivalence classes of  and . It depends in an essential way from the specific relative position that  and  occupy in a larger space .

One can define norms on  and  by

Equipped with these norms, the intersection and the sum are Banach spaces. The following inclusions are all continuous:

Interpolation studies the family of spaces  that are intermediate spaces between  and  in the sense that

where the two inclusions maps are continuous.

An example of this situation is the pair , where the two Banach spaces are continuously embedded in the space  of measurable functions on the real line, equipped with the topology of convergence in measure. In this situation, the spaces , for  are intermediate between  and . More generally,

with continuous injections, so that, under the given condition,  is intermediate between  and .

Definition. Given two compatible couples  and , an interpolation pair is a couple  of Banach spaces with the two following properties:
The space X is intermediate between  and , and Y is intermediate between  and .
If  is any linear operator from  to , which maps continuously  to  and  to , then it also maps continuously  to .

The interpolation pair  is said to be of exponent  (with ) if there exists a constant  such that

for all operators  as above. The notation  is for the norm of  as a map from  to . If , we say that  is an exact interpolation pair of exponent .

Complex interpolation 
If the scalars are complex numbers, properties of complex analytic functions are used to define an interpolation space. Given a compatible couple (X0, X1) of Banach spaces, the linear space  consists of all functions , that are analytic on  continuous on  and for which all the following subsets are bounded:

, 
, 
.

 is a Banach space under the norm

Definition. For , the complex interpolation space  is the linear subspace of  consisting of all values f(θ) when f varies in the preceding space of functions,

The norm on the complex interpolation space  is defined by

Equipped with this norm, the complex interpolation space  is a Banach space.

Theorem.  Given two compatible couples of Banach spaces  and , the pair  is an exact interpolation pair of exponent , i.e., if , is a linear operator bounded from  to , then  is bounded from  to  and 

The family of  spaces (consisting of complex valued functions) behaves well under complex interpolation.  If  is an arbitrary measure space, if  and , then

with equality of norms.  This fact is closely related to the Riesz–Thorin theorem.

Real interpolation 
There are two ways for introducing the real interpolation method. The first and most commonly used when actually identifying examples of interpolation spaces is the K-method. The second method, the J-method, gives the same interpolation spaces as the K-method when the parameter  is in . That the J- and K-methods agree is important for the study of duals of interpolation spaces: basically, the dual of an interpolation space constructed by the K-method appears to be a space constructed from the dual couple by the J-method; see below.

K-method 
The K-method of real interpolation can be used for Banach spaces over the field  of real numbers.

Definition. Let  be a compatible couple of Banach spaces. For  and every , let

Changing the order of the two spaces results in:

Let

The K-method of real interpolation consists in taking  to be the linear subspace of  consisting of all  such that .

Example 
An important example is that of the couple , where the functional  can be computed explicitly.  The measure  is supposed -finite. In this context, the best way of cutting the function  as sum of two functions  and  is, for some  to be chosen as function of , to let  be given for all  by

The optimal choice of  leads to the formula

where  is the decreasing rearrangement of .

J-method 
As with the K-method, the J-method can be used for real Banach spaces.

Definition. Let  be a compatible couple of Banach spaces. For  and for every vector , let 

A vector  in  belongs to the interpolation space  if and only if it can be written as

where  is measurable with values in  and such that

The norm of  in  is given by the formula

Relations between the interpolation methods 
The two real interpolation methods are equivalent when .

Theorem. Let  be a compatible couple of Banach spaces. If  and , then  with equivalence of norms.

The theorem covers degenerate cases that have not been excluded: for example if  and  form a direct sum, then the intersection and the J-spaces are the null space, and a simple computation shows that the K-spaces are also null.

When , one can speak, up to an equivalent renorming, about the Banach space obtained by the real interpolation method with parameters  and . The notation for this real interpolation space is . One has that

For a given value of , the real interpolation spaces increase with : if  and , the following continuous inclusion holds true:

Theorem. Given ,  and two compatible couples  and , the pair  is an exact interpolation pair of exponent .

A complex interpolation space is usually not isomorphic to one of the spaces given by the real interpolation method. However, there is a general relationship.

Theorem. Let  be a compatible couple of Banach spaces. If , then

Examples 
When  and , the space of continuously differentiable functions on , the  interpolation method, for , gives the Hölder space  of exponent .  This is because the K-functional  of this couple is equivalent to

Only values  are interesting here.

Real interpolation between  spaces gives the family of Lorentz spaces. Assuming  and , one has:

with equivalent norms. This follows from an inequality of Hardy and from the value given above of the K-functional for this compatible couple. When , the Lorentz space  is equal to , up to renorming. When , the Lorentz space  is equal to weak-.

The reiteration theorem 
An intermediate space  of the compatible couple  is said to be of class θ if 

with continuous injections. Beside all real interpolation spaces  with parameter  and , the complex interpolation space  is an intermediate space of class  of the compatible couple .

The reiteration theorems says, in essence, that interpolating with a parameter  behaves, in some way, like forming a convex combination : taking a further convex combination of two convex combinations gives another convex combination.

Theorem. Let  be intermediate spaces of the compatible couple , of class  and  respectively, with . When  and , one has 

It is notable that when interpolating with the real method between  and , only the values of  and  matter. Also,  and  can be complex interpolation spaces between  and , with parameters  and  respectively.

There is also a reiteration theorem for the complex method.

Theorem. Let  be a compatible couple of complex Banach spaces, and assume that  is dense in  and in . Let  and , where . Assume further that  is dense in . Then, for every , 

The density condition is always satisfied when  or .

Duality 
Let  be a compatible couple, and assume that  is dense in X0 and in X1. In this case, the restriction map from the (continuous) dual  of ,  to the dual of  is one-to-one. It follows that the pair of duals  is a compatible couple continuously embedded in the dual .

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

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

In general, the dual of the space  is equal to  a space defined by a variant of the complex method. The upper-θ and lower-θ methods do not coincide in general, but they do if at least one of X0, X1 is a reflexive space.

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

Theorem. Let  and  a compatible couple of real Banach spaces.  Assume that  is dense in  and in . Then  where

Discrete definitions 
Since the function  varies regularly (it is increasing, but  is decreasing), the definition of the -norm of a vector , previously given by an integral, is equivalent to a definition given by a series. This series is obtained by breaking  into pieces  of equal mass for the measure ,

In the special case where  is continuously embedded in , one can omit the part of the series with negative indices . In this case, each of the functions  defines an equivalent norm on .

The interpolation space  is a "diagonal subspace" of an -sum of a sequence of Banach spaces (each one being isomorphic to ). Therefore, when  is finite, the dual of  is a quotient of the -sum of the duals, , which leads to the following formula for the discrete -norm of a functional x in the dual of :

The usual formula for the discrete -norm is obtained by changing  to .

The discrete definition makes several questions easier to study, among which the already mentioned identification of the dual. Other such questions are compactness or weak-compactness of linear operators.  Lions and Peetre have proved that:Theorem. If the linear operator  is compact from  to a Banach space  and bounded from  to , then  is compact from  to  when , .

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.

 A general interpolation method 
The space  used for the discrete definition can be replaced by an arbitrary sequence space Y with unconditional basis, and the weights , , that are used for the -norm, can be replaced by general weights

The interpolation space  consists of the vectors  in  such that

where {yn} is the unconditional basis of . This abstract method can be used, for example, for the proof of 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,

These spaces are spaces of measurable functions on  when , and of tempered distributions on  when . For the rest of the section, the following setting and notation will be used:

Complex interpolation works well on the class of Sobolev spaces  (the Bessel potential spaces) as well as Besov spaces:

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

When  but , real interpolation between Sobolev spaces gives a Besov space:

Also,

 See also 
 Riesz–Thorin theorem
 Marcinkiewicz interpolation theorem

 Notes 

 References 
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Leoni, Giovanni (2017). A First Course in Sobolev Spaces: Second Edition. Graduate Studies in Mathematics. 181'. American Mathematical Society. pp. 734. .
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Banach spaces
Fourier analysis
Sobolev spaces