By Jöran Bergh, Jörgen Löfström (auth.)

The works of Jaak Peetre represent the most physique of this treatise. vital members also are J. L. Lions and A. P. Calderon, let alone numerous others. We, the current authors, have therefore purely compiled and defined the works of others (with the exception of some minor contributions of our own). allow us to point out the foundation of this treatise. a number of years in the past, J. Peetre recommended to the second one writer, J. Lofstrom, writing a ebook on interpolation thought and he so much generously positioned at Lofstrom's disposal an unfinished manu­ script, protecting elements of bankruptcy 1-3 and five of this publication. consequently, LOfstrom ready a primary tough, yet fairly entire manuscript of lecture notes. This was once then partially rewritten and thouroughly revised via the 1st writer, J. Bergh, who additionally ready the notes and remark and lots of the routines. during the paintings, we have now had the nice fortune of having fun with Jaak Peetre's variety patronage and valuable guidance. we wish to show our deep gratitude to him. thank you also are as a result of our colleagues for his or her aid and aid. eventually, we're truly thankful to Boe1 Engebrand, Lena Mattsson and Birgit Hoglund for his or her specialist typing of our manuscript.

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6. (Gustavsson [1]). , the norms are now only semidefinite. Moreover, let Ai c m:, i = 0,1, m: being a linear space. Put A={Ao,Ad and the null space of the couple A. Show that N(A) is a closed linear subspace of Ao+Aj equipped with the seminorm in the definition of N(A). lA, =0, i=O,1. 7. (Gagliardo [2]). Let A and B be (semi-)normed linear spaces and AcB. The Gagliardo completion of A relative to B, written A B,\ is the set of all bEB for which there exists a sequence (an) bounded in A and with the limit b in B.

K of all normed vector spaces. We assume that the mappings T: A-4B are all bounded linear operators from A to B. We let C(} I stand for a category of compatible couples A = (Ao, AI)' i. e. such that Ao and A I are compatible and such that Ao + A I and Ao n A I are spaces in C(}. The morphisms T: (A o,A 1 )-4(B o,B 1 ) in C(} I are all bounded linear mappings from Ao + Al to Bo + BI such that are morphisms in C(}. Here TA denotes the restriction of T to A. With the natural definitions of composite morphism and identity, it is easy to see that C(} I is in fact a category.

By Gagliardo [1], Peetre [17], Tartar [1], Brezis [1]. For additional references, see Peetre [17]. (Cf. ) "Non-linear" indicates that non-linear operators are admitted: e. , Lipschitz and Holder operators. Cf. 13. There are applications to partial differential equations: Tartar [1], Brezis [1]. 6. 9. 1-2. The functorial approach to interpolation merely provides a convenient framework for the underlying primitive ideas, and makes the exposition more stringent. 3. Introducing couples A=(Ao,Ai) we have assumed the existence of a Hausdorff topological vector space \ll, such that Ai c \ll (i = 0, 1).

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