{"history":[],"article":{"id":"56a63ff3-7cd4-44a5-b323-979b652e3aa5","title":"Interpolation space","slug":"interpolation-space","content":":::summary\n:::\n\n# \nIn the field of [mathematical analysis](https://bliptext.com/articles/mathematical-analysis), an **interpolation space** is a space which lies \"in between\" two other [Banach spaces](https://bliptext.com/articles/banach-space).\nThe main applications are in [Sobolev spaces](https://bliptext.com/articles/sobolev-space), where spaces of functions that have a noninteger number of [derivatives](https://bliptext.com/articles/derivative) are interpolated from the spaces of functions with integer number of [derivatives](https://bliptext.com/articles/derivative).\n\n# History\nThe theory of interpolation of vector spaces began by an observation of [Józef Marcinkiewicz](https://bliptext.com/articles/j-zef-marcinkiewicz), later generalized and now known as the [Riesz-Thorin theorem](https://bliptext.com/articles/riesz-thorin-theorem).\nIn simple terms, if a linear function is continuous on a certain [[Lp space|space\nLp\n]] and also on a certain space\nLq\n, then it is also continuous on the space\nLr\n, for any intermediate r between p and q. In other words,\nLr\nis a space which is intermediate between\nLp\nand\nLq\n.\nIn the development of Sobolev spaces, it became clear that the [trace](https://bliptext.com/articles/trace-operator) spaces were not any of the usual function spaces (with integer number of derivatives), and [Jacques-Louis Lions](https://bliptext.com/articles/jacques-louis-lions) discovered that indeed these [trace](https://bliptext.com/articles/trace-operator) spaces were constituted of functions that have a noninteger degree of differentiability.\nMany methods were designed to generate such spaces of functions, including the [Fourier transform](https://bliptext.com/articles/fourier-transform), complex interpolation,\nreal interpolation,\nas well as other tools (see e.g. [fractional derivative](https://bliptext.com/articles/fractional-derivative)).\n\n# The setting of interpolation\nA [Banach space](https://bliptext.com/articles/banach-space) X is said to be continuously embedded in a [Hausdorff](https://bliptext.com/articles/hausdorff-space) [topological vector space](https://bliptext.com/articles/topological-vector-space) Z when X is a linear subspace of Z such that the inclusion map from X into Z is continuous.\nA **compatible couple**\n(X0, X1)\nof Banach spaces consists of two Banach spaces\nX0\nand\nX1\nthat are continuously embedded in the same Hausdorff topological vector space Z. The embedding in a linear space Z allows to consider the two linear subspaces\nand\nInterpolation does not depend only upon the isomorphic (nor isometric) equivalence classes of\nX0\nand\nX1\n. It depends in an essential way from the specific relative position that\nX0\nand\nX1\noccupy in a larger space Z.\nOne can define norms on\nX0 ∩ X1\nand\nX0 + X1\nby\nEquipped with these norms, the intersection and the sum are Banach spaces.\nThe following inclusions are all continuous:\nInterpolation studies the family of spaces X that are **intermediate spaces** between\nX0\nand\nX1\nin the sense that\nwhere the two inclusions maps are continuous.\nAn example of this situation is the pair\n(L1(R), L∞(R))\n, where the two Banach spaces are continuously embedded in the space Z of measurable functions on the real line, equipped with the topology of convergence in measure.\nIn this situation, the spaces\nLp(R)\n, for\n1 ≤ p ≤ ∞\nare intermediate between\nL1(R)\nand\nL∞(R)\n. More generally,\nwith continuous injections, so that, under the given condition,\nLp(R)\nis intermediate between\nLp0(R)\nand\nLp1(R)\n.\n(X0, X1)\nand\n(Y0, Y1)\n, an **interpolation pair** is a couple\n(X, Y)\nof Banach spaces with the two following properties:\nX0\nand\nX1\n, and Y is intermediate between\nY0\nand\nY1\n.\nL\nis any linear operator from\nX0 + X1\nto\nY0 + Y1\n, which maps continuously\nX0\nto\nY0\nand\nX1\nto\nY1\n, then it also maps continuously\nX\nto\nY\n.\nThe interpolation pair\n(X, Y)\nis said to be of **exponent θ** (with\n0 < θ < 1\n) if there exists a constant\nC\nsuch that\nfor all operators L as above.\nThe notation\nLX,Y\nis for the norm of\nL\nas a map from\nX\nto\nY\n. If\n, we say that\n(X, Y)\nis an **exact interpolation pair of exponent θ**.\n\n# Complex interpolation\nIf the scalars are [complex numbers](https://bliptext.com/articles/complex-number), properties of complex [analytic functions](https://bliptext.com/articles/analytic-function) are used to define an interpolation space.\nGiven a compatible couple (X0, X1) of Banach spaces, the linear space consists of all functions\n f : C → X0 + X1\n, that are analytic on\ncontinuous on\nand for which all the following subsets are bounded:\n{ f (z) : z ∈ S} ⊂ X0 + X1\n,\n{ f (it) : t ∈ R} ⊂ X0\n,\n{ f (1 + it) : t ∈ R} ⊂ X1\n.\nis a Banach space under the norm\n**Definition.** For\n0 < θ < 1\n, the **complex interpolation space**\n(X0, X1)θ\nis the linear subspace of\nX0 + X1\nconsisting of all values f(θ) when f varies in the preceding space of functions,\nThe norm on the complex interpolation space\n(X0, X1)θ\nis defined by\nEquipped with this norm, the complex interpolation space\n(X0, X1)θ\nis a Banach space.\n(X0, X1)\nand\n(Y0, Y1)\n, the pair\n((X0, X1)θ, (Y0, Y1)θ)\nis an exact interpolation pair of exponent θ, i.e., if\nT : X0 + X1 → Y0 + Y1\n, is a linear operator bounded from\nXj\nto\nYj, j = 0, 1\n, then T is bounded from\n(X0, X1)θ\nto\n(Y0, Y1)θ\nand\nThe family of\nLp\nspaces (consisting of complex valued functions) behaves well under complex interpolation.\nIf\n(R, Σ, μ)\nis an arbitrary [measure space](https://bliptext.com/articles/measure-mathematics), if\n1 ≤ p0, p1 ≤ ∞\nand\n0 < θ < 1\n, then\nwith equality of norms.\nThis fact is closely related to the [Riesz–Thorin theorem](https://bliptext.com/articles/riesz-thorin-theorem).\n\n# Real interpolation\nThere are two ways for introducing the **real interpolation method**.\nThe first and most commonly used when actually identifying examples of interpolation spaces is the K-method.\nThe second method, the J-method, gives the same interpolation spaces as the K-method when the parameter θ is in\n(0, 1)\n. 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](https://bliptext.com/articles/interpolation-space).\n\n## K-method\nThe K-method of real interpolation can be used for Banach spaces over the field\nR\nof [real numbers](https://bliptext.com/articles/real-number).\n**Definition.** Let\n(X0, X1)\nbe a compatible couple of Banach spaces.\nFor\nt > 0\nand every\nx ∈ X0 + X1\n, let\nChanging the order of the two spaces results in:\nLet\nThe K-method of real interpolation consists in taking\nKθ,q(X0, X1)\nto be the linear subspace of\nX0 + X1\nconsisting of all x such that\nxθ,q;K < ∞\n.\n\n### Example\nAn important example is that of the couple\n(L1(R, Σ, μ), L∞(R, Σ, μ))\n, where the functional\nK(t, f ; L1, L∞)\ncan be computed explicitly.\nThe measure μ is supposed [σ-finite](https://bliptext.com/articles/finite-measure).\nIn this context, the best way of cutting the function\n f ∈ L1 + L∞\nas sum of two functions\n f0 ∈ L1 \nand\n f1 ∈ L∞ \nis, for some\ns > 0\nto be chosen as function of t, to let\n f1(x)\nbe given for all\nx ∈ R\nby\nThe optimal choice of s leads to the formula\nwhere\n f∗\nis the [decreasing rearrangement](https://bliptext.com/articles/lorentz-space) of\n f \n.\n\n## J-method\nAs with the K-method, the J-method can be used for real Banach spaces.\n**Definition.** Let\n(X0, X1)\nbe a compatible couple of Banach spaces.\nFor\nt > 0\nand for every vector\nx ∈ X0 ∩ X1\n, let\nA vector x in\nX0 + X1\nbelongs to the interpolation space\nJθ,q(X0, X1)\nif and only if it can be written as\nwhere\nv(t)\nis measurable with values in\nX0 ∩ X1\nand such that\nThe norm of x in\nJθ,q(X0, X1)\nis given by the formula\n\n## Relations between the interpolation methods\nThe two real interpolation methods are equivalent when\n0 < θ < 1\n.\n(X0, X1)\nbe a compatible couple of Banach spaces.\nIf\n0 < θ < 1\nand\n1 ≤ q ≤ ∞\n, then with [equivalence of norms](https://bliptext.com/articles/norm-mathematics).\nThe theorem covers degenerate cases that have not been excluded: for example if\nX0\nand\nX1\nform 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.\nWhen\n0 < θ < 1\n, one can speak, up to an equivalent renorming, about the Banach space obtained by the real interpolation method with parameters θ and q. The notation for this real interpolation space is\n(X0, X1)θ,q\n. One has that\nFor a given value of θ, the real interpolation spaces increase with q: if\n0 < θ < 1\nand\n1 ≤ q ≤ r ≤ ∞\n, the following continuous inclusion holds true:\n0 < θ < 1\n,\n1 ≤ q ≤ ∞\nand two compatible couples\n(X0, X1)\nand\n(Y0, Y1)\n, the pair\n((X0, X1)θ,q, (Y0, Y1)θ,q)\nis an exact interpolation pair of exponent θ.\nA complex interpolation space is usually not isomorphic to one of the spaces given by the real interpolation method.\nHowever, there is a general relationship.\n(X0, X1)\nbe a compatible couple of Banach spaces.\nIf\n0 < θ < 1\n, then\n\n### Examples\nWhen\nand\n, the space of continuously differentiable functions on\n[0, 1]\n, the\n(θ, ∞)\ninterpolation method, for\n0 < θ < 1\n, gives the [Hölder space](https://bliptext.com/articles/h-lder-condition)\nC0,θ\nof exponent θ.\nThis is because the K-functional\nK(f, t; X0, X1)\nof this couple is equivalent to\nOnly values\n0 < t < 1\nare interesting here.\nReal interpolation between\nLp\nspaces gives the family of [Lorentz spaces](https://bliptext.com/articles/lorentz-space).\nAssuming\n0 < θ < 1\nand\n1 ≤ q ≤ ∞\n, one has:\nwith equivalent norms.\nThis follows from an [inequality of Hardy](https://bliptext.com/articles/hardy-s-inequality) and from the value given above of the K-functional for this compatible couple.\nWhen\n, the Lorentz space\nLp,p\nis equal to\nLp\n, up to renorming.\nWhen\n, the Lorentz space\nLp,∞\nis equal to [[Lp space#Weak Lp|weak-\nLp\n]].\n\n# The reiteration theorem\nAn intermediate space X of the compatible couple\n(X0, X1)\nis said to be of class θ if\nwith continuous injections.\nBeside all real interpolation spaces\n(X0, X1)θ,q\nwith parameter θ and\n1 ≤ q ≤ ∞\n, the complex interpolation space\n(X0, X1)θ\nis an intermediate space of class θ of the compatible couple\n(X0, X1)\n.\nThe reiteration theorems says, in essence, that interpolating with a parameter θ behaves, in some way, like forming a [convex combination](https://bliptext.com/articles/convex-combination)\nA0, A1\nbe intermediate spaces of the compatible couple\n(X0, X1)\n, of class\nθ0\nand\nθ1\nrespectively, with\n0 < θ0 ≠ θ1 < 1\n. When\n0 < θ < 1\nand\n1 ≤ q ≤ ∞\n, one has\nIt is notable that when interpolating with the real method between\nand\n, only the values of\nθ0\nand\nθ1\nmatter.\nAlso,\nA0\nand\nA1\ncan be complex interpolation spaces between\nX0\nand\nX1\n, with parameters\nθ0\nand\nθ1\nrespectively.\nThere is also a reiteration theorem for the complex method.\n(X0, X1)\nbe a compatible couple of complex Banach spaces, and assume that\nX0 ∩ X1\nis dense in\nX0\nand in\nX1\n. Let\nand\n, where\n0 ≤ θ0 ≤ θ1 ≤ 1\n. Assume further that\nX0 ∩ X1\nis dense in\nA0 ∩ A1\n. Then, for every\n0 ≤ θ ≤ 1\n,\nThe density condition is always satisfied when\nX0 ⊂ X1\nor\nX1 ⊂ X0\n.\n\n# Duality\nLet\n(X0, X1)\nbe a compatible couple, and assume that\nX0 ∩ X1\nis dense in X0 and in X1.\nIn this case, the restriction map from the (continuous) [dual](https://bliptext.com/articles/dual-space) X'_j of\nXj\n,\nto the dual of\nX0 ∩ X1\nis one-to-one.\nIt follows that the pair of duals is a compatible couple continuously embedded in the dual\n(X0 ∩ X1)′\n.\nFor the complex interpolation method, the following duality result holds:\n(X0, X1)\nbe a compatible couple of complex Banach spaces, and assume that\nX0 ∩ X1\nis dense in\nX0\nand in\nX1\n. If\nX0\nand\nX1\nare [reflexive](https://bliptext.com/articles/reflexive-space), then the dual of the complex interpolation space is obtained by interpolating the duals,\nIn general, the dual of the space\n(X0, X1)θ\nis equal to a space defined by a variant of the complex method.\nThe 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.\nFor the real interpolation method, the duality holds provided that the parameter q is finite:\n0 < θ < 1, 1 ≤ q < ∞\nand\n(X0, X1)\na compatible couple of real Banach spaces.\nAssume that\nX0 ∩ X1\nis dense in\nX0\nand in\nX1\n. Then where\n\n# Discrete definitions\nSince the function\nt → K(x, t)\nvaries regularly (it is increasing, but\n1⁄tK(x, t)\nis decreasing), the definition of the\nKθ,q\n-norm of a vector n, previously given by an integral, is equivalent to a definition given by a series.\nThis series is obtained by breaking\n(0, ∞)\ninto pieces\n(2n, 2n+1)\nof equal mass for the measure\ndt⁄t\n,\nIn the special case where\nX0\nis continuously embedded in\nX1\n, one can omit the part of the series with negative indices n. In this case, each of the functions\nx → K(x, 2n; X0, X1)\ndefines an equivalent norm on\nX1\n.\nThe interpolation space\n(X0, X1)θ,q\nis a \"diagonal subspace\" of an\nℓ q\n-sum of a sequence of Banach spaces (each one being isomorphic to\nX0 + X1\n). Therefore, when q is finite, the dual of\n(X0, X1)θ,q\nis a [quotient](https://bliptext.com/articles/banach-space) of the\nℓ p\n-sum of the duals,\n1⁄p + 1⁄q = 1\n, which leads to the following formula for the discrete\nJθ,p\n-norm of a functional x' in the dual of\n(X0, X1)θ,q\nThe usual formula for the discrete\nJθ,p\n-norm is obtained by changing n to\n−n\n.\nThe discrete definition makes several questions easier to study, among which the already mentioned identification of the dual.\nOther such questions are compactness or weak-compactness of linear operators.\nLions and Peetre have proved that:\nX0\nto a Banach space Y and bounded from\nX1\nto Y, then T is compact from\n(X0, X1)θ,q\nto Y when\n0 < θ < 1\n,\n1 ≤ q ≤ ∞\n.\nDavis, Figiel, Johnson and Pełczyński have used interpolation in their proof of the following result:\n\n## A general interpolation method\nThe space\nℓ q\nused for the discrete definition can be replaced by an arbitrary [sequence space](https://bliptext.com/articles/sequence-space) Y with [unconditional basis](https://bliptext.com/articles/schauder-basis), and the weights\n,\n, that are used for the\nKθ,q\n-norm, can be replaced by general weights\nThe interpolation space\nK(X0, X1, Y, {an}, {bn})\nconsists of the vectors x in\nX0 + X1\nsuch that\nwhere {yn} is the unconditional basis of Y. This abstract method can be used, for example, for the proof of the following result:\n**Theorem.** A Banach space with unconditional basis is isomorphic to a complemented subspace of a space with [symmetric basis](https://bliptext.com/articles/schauder-basis).\n\n# Interpolation of Sobolev and Besov spaces\nSeveral interpolation results are available for [Sobolev spaces](https://bliptext.com/articles/sobolev-space) and [Besov spaces](https://bliptext.com/articles/besov-space) on **R**n,\nThese spaces are spaces of [measurable functions](https://bliptext.com/articles/measurable-function) on\nRn\nwhen\ns ≥ 0\n, and of [tempered distributions](https://bliptext.com/articles/distribution-mathematics) on\nRn\nwhen\ns < 0\n. For the rest of the section, the following setting and notation will be used:\nComplex interpolation works well on the class of Sobolev spaces H^{s}_{p} (the [Bessel potential spaces](https://bliptext.com/articles/sobolev-space)) as well as Besov spaces:\nReal interpolation between Sobolev spaces may give Besov spaces, except when\n,\nWhen\ns0 ≠ s1\nbut\n, real interpolation between Sobolev spaces gives a Besov space:\nAlso,","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T10:57:44.851Z","updated_at":"2025-01-26T10:57:44.851Z","search_vector":"'interpol':1 'space':2","search_content":" \n \n\n# \nIn the field of [mathematical analysis](https://bliptext.com/articles/mathematical-analysis), an **interpolation space** is a space which lies \"in between\" two other [Banach spaces](https://bliptext.com/articles/banach-space).\nThe main applications are in [Sobolev spaces](https://bliptext.com/articles/sobolev-space), where spaces of functions that have a noninteger number of [derivatives](https://bliptext.com/articles/derivative) are interpolated from the spaces of functions with integer number of [derivatives](https://bliptext.com/articles/derivative).\n\n# History\nThe theory of interpolation of vector spaces began by an observation of [Józef Marcinkiewicz](https://bliptext.com/articles/j-zef-marcinkiewicz), later generalized and now known as the [Riesz-Thorin theorem](https://bliptext.com/articles/riesz-thorin-theorem).\nIn simple terms, if a linear function is continuous on a certain [[Lp space|space\nLp\n]] and also on a certain space\nLq\n, then it is also continuous on the space\nLr\n, for any intermediate r between p and q. In other words,\nLr\nis a space which is intermediate between\nLp\nand\nLq\n.\nIn the development of Sobolev spaces, it became clear that the [trace](https://bliptext.com/articles/trace-operator) spaces were not any of the usual function spaces (with integer number of derivatives), and [Jacques-Louis Lions](https://bliptext.com/articles/jacques-louis-lions) discovered that indeed these [trace](https://bliptext.com/articles/trace-operator) spaces were constituted of functions that have a noninteger degree of differentiability.\nMany methods were designed to generate such spaces of functions, including the [Fourier transform](https://bliptext.com/articles/fourier-transform), complex interpolation,\nreal interpolation,\nas well as other tools (see e.g. [fractional derivative](https://bliptext.com/articles/fractional-derivative)).\n\n# The setting of interpolation\nA [Banach space](https://bliptext.com/articles/banach-space) X is said to be continuously embedded in a [Hausdorff](https://bliptext.com/articles/hausdorff-space) [topological vector space](https://bliptext.com/articles/topological-vector-space) Z when X is a linear subspace of Z such that the inclusion map from X into Z is continuous.\nA **compatible couple**\n(X0, X1)\nof Banach spaces consists of two Banach spaces\nX0\nand\nX1\nthat are continuously embedded in the same Hausdorff topological vector space Z. The embedding in a linear space Z allows to consider the two linear subspaces\nand\nInterpolation does not depend only upon the isomorphic (nor isometric) equivalence classes of\nX0\nand\nX1\n. It depends in an essential way from the specific relative position that\nX0\nand\nX1\noccupy in a larger space Z.\nOne can define norms on\nX0 ∩ X1\nand\nX0 + X1\nby\nEquipped with these norms, the intersection and the sum are Banach spaces.\nThe following inclusions are all continuous:\nInterpolation studies the family of spaces X that are **intermediate spaces** between\nX0\nand\nX1\nin the sense that\nwhere the two inclusions maps are continuous.\nAn example of this situation is the pair\n(L1(R), L∞(R))\n, where the two Banach spaces are continuously embedded in the space Z of measurable functions on the real line, equipped with the topology of convergence in measure.\nIn this situation, the spaces\nLp(R)\n, for\n1 ≤ p ≤ ∞\nare intermediate between\nL1(R)\nand\nL∞(R)\n. More generally,\nwith continuous injections, so that, under the given condition,\nLp(R)\nis intermediate between\nLp0(R)\nand\nLp1(R)\n.\n(X0, X1)\nand\n(Y0, Y1)\n, an **interpolation pair** is a couple\n(X, Y)\nof Banach spaces with the two following properties:\nX0\nand\nX1\n, and Y is intermediate between\nY0\nand\nY1\n.\nL\nis any linear operator from\nX0 + X1\nto\nY0 + Y1\n, which maps continuously\nX0\nto\nY0\nand\nX1\nto\nY1\n, then it also maps continuously\nX\nto\nY\n.\nThe interpolation pair\n(X, Y)\nis said to be of **exponent θ** (with\n0 0\nand every\nx ∈ X0 + X1\n, let\nChanging the order of the two spaces results in:\nLet\nThe K-method of real interpolation consists in taking\nKθ,q(X0, X1)\nto be the linear subspace of\nX0 + X1\nconsisting of all x such that\nxθ,q;K 0\nto be chosen as function of t, to let\n f1(x)\nbe given for all\nx ∈ R\nby\nThe optimal choice of s leads to the formula\nwhere\n f∗\nis the [decreasing rearrangement](https://bliptext.com/articles/lorentz-space) of\n f \n.\n\n## J-method\nAs with the K-method, the J-method can be used for real Banach spaces.\n**Definition.** Let\n(X0, X1)\nbe a compatible couple of Banach spaces.\nFor\nt > 0\nand for every vector\nx ∈ X0 ∩ X1\n, let\nA vector x in\nX0 + X1\nbelongs to the interpolation space\nJθ,q(X0, X1)\nif and only if it can be written as\nwhere\nv(t)\nis measurable with values in\nX0 ∩ X1\nand such that\nThe norm of x in\nJθ,q(X0, X1)\nis given by the formula\n\n## Relations between the interpolation methods\nThe two real interpolation methods are equivalent when\n0 < θ < 1\n.\n(X0, X1)\nbe a compatible couple of Banach spaces.\nIf\n0 < θ < 1\nand\n1 ≤ q ≤ ∞\n, then with [equivalence of norms](https://bliptext.com/articles/norm-mathematics).\nThe theorem covers degenerate cases that have not been excluded: for example if\nX0\nand\nX1\nform 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.\nWhen\n0 < θ < 1\n, one can speak, up to an equivalent renorming, about the Banach space obtained by the real interpolation method with parameters θ and q. The notation for this real interpolation space is\n(X0, X1)θ,q\n. One has that\nFor a given value of θ, the real interpolation spaces increase with q: if\n0 < θ < 1\nand\n1 ≤ q ≤ r ≤ ∞\n, the following continuous inclusion holds true:\n0 < θ < 1\n,\n1 ≤ q ≤ ∞\nand two compatible couples\n(X0, X1)\nand\n(Y0, Y1)\n, the pair\n((X0, X1)θ,q, (Y0, Y1)θ,q)\nis an exact interpolation pair of exponent θ.\nA complex interpolation space is usually not isomorphic to one of the spaces given by the real interpolation method.\nHowever, there is a general relationship.\n(X0, X1)\nbe a compatible couple of Banach spaces.\nIf\n0 < θ < 1\n, then\n\n### Examples\nWhen\nand\n, the space of continuously differentiable functions on\n[0, 1]\n, the\n(θ, ∞)\ninterpolation method, for\n0 < θ < 1\n, gives the [Hölder space](https://bliptext.com/articles/h-lder-condition)\nC0,θ\nof exponent θ.\nThis is because the K-functional\nK(f, t; X0, X1)\nof this couple is equivalent to\nOnly values\n0 < t < 1\nare interesting here.\nReal interpolation between\nLp\nspaces gives the family of [Lorentz spaces](https://bliptext.com/articles/lorentz-space).\nAssuming\n0 < θ < 1\nand\n1 ≤ q ≤ ∞\n, one has:\nwith equivalent norms.\nThis follows from an [inequality of Hardy](https://bliptext.com/articles/hardy-s-inequality) and from the value given above of the K-functional for this compatible couple.\nWhen\n, the Lorentz space\nLp,p\nis equal to\nLp\n, up to renorming.\nWhen\n, the Lorentz space\nLp,∞\nis equal to [[Lp space#Weak Lp|weak-\nLp\n]].\n\n# The reiteration theorem\nAn intermediate space X of the compatible couple\n(X0, X1)\nis said to be of class θ if\nwith continuous injections.\nBeside all real interpolation spaces\n(X0, X1)θ,q\nwith parameter θ and\n1 ≤ q ≤ ∞\n, the complex interpolation space\n(X0, X1)θ\nis an intermediate space of class θ of the compatible couple\n(X0, X1)\n.\nThe reiteration theorems says, in essence, that interpolating with a parameter θ behaves, in some way, like forming a [convex combination](https://bliptext.com/articles/convex-combination)\nA0, A1\nbe intermediate spaces of the compatible couple\n(X0, X1)\n, of class\nθ0\nand\nθ1\nrespectively, with\n0 < θ0 ≠ θ1 < 1\n. When\n0 < θ < 1\nand\n1 ≤ q ≤ ∞\n, one has\nIt is notable that when interpolating with the real method between\nand\n, only the values of\nθ0\nand\nθ1\nmatter.\nAlso,\nA0\nand\nA1\ncan be complex interpolation spaces between\nX0\nand\nX1\n, with parameters\nθ0\nand\nθ1\nrespectively.\nThere is also a reiteration theorem for the complex method.\n(X0, X1)\nbe a compatible couple of complex Banach spaces, and assume that\nX0 ∩ X1\nis dense in\nX0\nand in\nX1\n. Let\nand\n, where\n0 ≤ θ0 ≤ θ1 ≤ 1\n. Assume further that\nX0 ∩ X1\nis dense in\nA0 ∩ A1\n. Then, for every\n0 ≤ θ ≤ 1\n,\nThe density condition is always satisfied when\nX0 ⊂ X1\nor\nX1 ⊂ X0\n.\n\n# Duality\nLet\n(X0, X1)\nbe a compatible couple, and assume that\nX0 ∩ X1\nis dense in X0 and in X1.\nIn this case, the restriction map from the (continuous) [dual](https://bliptext.com/articles/dual-space) X'_j of\nXj\n,\nto the dual of\nX0 ∩ X1\nis one-to-one.\nIt follows that the pair of duals is a compatible couple continuously embedded in the dual\n(X0 ∩ X1)′\n.\nFor the complex interpolation method, the following duality result holds:\n(X0, X1)\nbe a compatible couple of complex Banach spaces, and assume that\nX0 ∩ X1\nis dense in\nX0\nand in\nX1\n. If\nX0\nand\nX1\nare [reflexive](https://bliptext.com/articles/reflexive-space), then the dual of the complex interpolation space is obtained by interpolating the duals,\nIn general, the dual of the space\n(X0, X1)θ\nis equal to a space defined by a variant of the complex method.\nThe 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.\nFor the real interpolation method, the duality holds provided that the parameter q is finite:\n0 < θ < 1, 1 ≤ q < ∞\nand\n(X0, X1)\na compatible couple of real Banach spaces.\nAssume that\nX0 ∩ X1\nis dense in\nX0\nand in\nX1\n. Then where\n\n# Discrete definitions\nSince the function\nt → K(x, t)\nvaries regularly (it is increasing, but\n1⁄tK(x, t)\nis decreasing), the definition of the\nKθ,q\n-norm of a vector n, previously given by an integral, is equivalent to a definition given by a series.\nThis series is obtained by breaking\n(0, ∞)\ninto pieces\n(2n, 2n+1)\nof equal mass for the measure\ndt⁄t\n,\nIn the special case where\nX0\nis continuously embedded in\nX1\n, one can omit the part of the series with negative indices n. In this case, each of the functions\nx → K(x, 2n; X0, X1)\ndefines an equivalent norm on\nX1\n.\nThe interpolation space\n(X0, X1)θ,q\nis a \"diagonal subspace\" of an\nℓ q\n-sum of a sequence of Banach spaces (each one being isomorphic to\nX0 + X1\n). Therefore, when q is finite, the dual of\n(X0, X1)θ,q\nis a [quotient](https://bliptext.com/articles/banach-space) of the\nℓ p\n-sum of the duals,\n1⁄p + 1⁄q = 1\n, which leads to the following formula for the discrete\nJθ,p\n-norm of a functional x' in the dual of\n(X0, X1)θ,q\nThe usual formula for the discrete\nJθ,p\n-norm is obtained by changing n to\n−n\n.\nThe discrete definition makes several questions easier to study, among which the already mentioned identification of the dual.\nOther such questions are compactness or weak-compactness of linear operators.\nLions and Peetre have proved that:\nX0\nto a Banach space Y and bounded from\nX1\nto Y, then T is compact from\n(X0, X1)θ,q\nto Y when\n0 < θ < 1\n,\n1 ≤ q ≤ ∞\n.\nDavis, Figiel, Johnson and Pełczyński have used interpolation in their proof of the following result:\n\n## A general interpolation method\nThe space\nℓ q\nused for the discrete definition can be replaced by an arbitrary [sequence space](https://bliptext.com/articles/sequence-space) Y with [unconditional basis](https://bliptext.com/articles/schauder-basis), and the weights\n,\n, that are used for the\nKθ,q\n-norm, can be replaced by general weights\nThe interpolation space\nK(X0, X1, Y, {an}, {bn})\nconsists of the vectors x in\nX0 + X1\nsuch that\nwhere {yn} is the unconditional basis of Y. This abstract method can be used, for example, for the proof of the following result:\n**Theorem.** A Banach space with unconditional basis is isomorphic to a complemented subspace of a space with [symmetric basis](https://bliptext.com/articles/schauder-basis).\n\n# Interpolation of Sobolev and Besov spaces\nSeveral interpolation results are available for [Sobolev spaces](https://bliptext.com/articles/sobolev-space) and [Besov spaces](https://bliptext.com/articles/besov-space) on **R**n,\nThese spaces are spaces of [measurable functions](https://bliptext.com/articles/measurable-function) on\nRn\nwhen\ns ≥ 0\n, and of [tempered distributions](https://bliptext.com/articles/distribution-mathematics) on\nRn\nwhen\ns < 0\n. For the rest of the section, the following setting and notation will be used:\nComplex interpolation works well on the class of Sobolev spaces H^{s}_{p} (the [Bessel potential spaces](https://bliptext.com/articles/sobolev-space)) as well as Besov spaces:\nReal interpolation between Sobolev spaces may give Besov spaces, except when\n,\nWhen\ns0 ≠ s1\nbut\n, real interpolation between Sobolev spaces gives a Besov space:\nAlso,","revisionCount":0},"hasMore":false,"nextCursor":null}