: Interpolation Spaces (Grundlehren der mathematischen Wissenschaften) (): J. Bergh, J. Lofstrom: Books. Citation. Bennett, Colin. Review: Jöran Bergh and Jörgen Löfström, Interpolation spaces, an introduction. Bull. Amer. Math. Soc. 84 (), no. 1, Interpolation spaces: an introduction. Front Cover. Jöran Bergh, Jörgen Löfström. Springer, – Mathematics – pages.

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## Interpolation space

Functional Analysis In the field of mathematical analysisan interpolation space is a space which lies “in between” two other Banach spaces. 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 derivativesand Jacques-Louis Lions discovered that indeed these trace spaces were constituted of functions that have a noninteger degree of differentiability. Let X 0X 1 be a compatible couple of Banach spaces.

Several interpolation results are available for Sobolev spaces and Besov spaces on R n[26]. Details Collect From This abstract method can be used, for example, for the proof of the following result:. See what’s been added to the collection in the current 1 2 3 4 5 6 weeks months years. The family of L p spaces consisting of complex valued functions behaves well under complex interpolation.

A Banach space X is said to be continuously embedded in a Hausdorff topological vector space Z when X is a linear subspace of Z such that the inclusion map from X into Z is continuous.

From Wikipedia, the free encyclopedia. In the Library Request this item to view in the Library’s reading rooms using your library card. In simple terms, if a linear function is continuous on a certain space L p and also on a certain space L qthen it is also continuous on the space L rfor any intermediate r between p and q.

The first and most commonly used when actually identifying examples of interpolation spaces is the K-method.

Interpolation studies the family of spaces X that are intermediate spaces between X 0 and X 1 in the sense that. Interpolation does not depend only upon the isomorphic nor isometric equivalence classes of X 0 and X 1. Retrieved from ” https: For the rest of the section, the following setting and notation will be used:.

Can I view this online? Many methods were designed to generate such spaces of functions, including the Fourier transformcomplex interpolation, [1] real interpolation, [2] as well as other tools see e.

Views Read Edit View history. Collection delivery service resumes on Wednesday 2 January That the J- and K-methods agree is important for the study of duals of interpolation spaces: It depends in an essential way from the specific relative position that X 0 and X 1 occupy in a larger space Z.

Equipped with these norms, the intersection and the sum are Banach spaces. 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. The K-method of real interpolation [7] can be used for Banach spaces over the field R of real numbers.

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The discrete definition makes spwces questions easier to study, among which the already mentioned identification of the dual. Other such questions are compactness or weak-compactness of linear operators. National Library of Australia. To learn more about Copies Direct watch this short online video. In other words, L r is a space which is intermediate between L p interpolatiin L q. The optimal choice of s leads to the formula [9].

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### Interpolation space of mixed Lorentz-Lebesgue spaces – MathOverflow

Further information on the Library’s opening hours is available at: Catalogue Persistent Identifier https: Browse titles authors subjects uniform titles series callnumbers dewey numbers starting from optional. This follows from an inequality of Hardy and from the value given above of the K-functional for this compatible couple.

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This fact is closely related to the Riesz—Thorin psaces. There are two ways for introducing the real interpolation method.

Can I borrow this item? How do I find a book? Paris in French From 25 December to 1 Januarythe Library’s Reading Rooms will be closed and no collection requests will be filled.

You must be logged in to Tag Records. A compatible couple X 0X 1 of Banach spaces consists of two Banach spaces X 0 and X 1 that are continuously embedded in the same Hausdorff topological vector space Z. In the special case where X 0 is continuously embedded in X 1one can omit the part of the series with negative indices n.