By Jöran Bergh, Jörgen Löfström (auth.)

The works of Jaak Peetre represent the most physique of this treatise. very important participants also are J. L. Lions and A. P. Calderon, let alone a number of others. We, the current authors, have therefore in basic terms compiled and defined the works of others (with the exception of some minor contributions of our own). allow us to point out the beginning of this treatise. a number of years in the past, J. Peetre steered to the second one writer, J. Lofstrom, writing a booklet on interpolation thought and he such a lot generously positioned at Lofstrom's disposal an unfinished manu script, masking elements of bankruptcy 1-3 and five of this booklet. consequently, LOfstrom ready a primary tough, yet fairly whole manuscript of lecture notes. This was once then partially rewritten and thouroughly revised through the 1st writer, J. Bergh, who additionally ready the notes and remark and many of the workouts. during the paintings, now we have had the nice fortune of having fun with Jaak Peetre's type patronage and helpful assistance. we wish to exhibit our deep gratitude to him. thank you also are as a result of our colleagues for his or her aid and support. ultimately, we're clearly thankful to Boe1 Engebrand, Lena Mattsson and Birgit Hoglund for his or her professional typing of our manuscript.

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**Sample text**

The existence of f follows from the Hahn-Banach theorem. If xEAi we have i = 0,1. Hence, since A and B are uniform interpolation spaces, I Tx II B ~ Cli x II A' xEA. Putting x=a we have IlbIIB~CllaIIA since Ta=b. Finally, if A and B are exact, obviously C = 1. The proof is complete. 7. A Duality Theorem Considering the category fJl of all Banach spaces we have the following. 1. Theorem. 4) is dense in both Ao and A1. 4'=(A~,A~) and A' denotes the dual of A. 4),. Proof: We prove only the first formula.

Suppose that X is an intermediate space with respect to A. S;Ct-oJ(t,a;A), aEJ(A). S; 1. 4). 5. 1. If A =(A o, AI) we have that Ao is of class CC(O; A) and AI is of class CC(1; A). This follows at once from the definition of J(t, a; A) and from the inequality It is sometimes convenient to write down the definition without explicit use of K(t, a) and J(t, a). Indeed, it is obvious that (a) Xis of class CCK (8;A) if and only if for any t>O there exist aoEAo and aIEA I , such that a=aO+a l and IlaollAo~CtOllallx and IIalll A1 ~CtO-lllallx' We can also show that (b) X is of class CCA8; A) if and only if we have In fact, if X is of class CCA8; A) we have that for all t>O.

Note that F is exact if we can take C = 1. The simplest interpolation functors are the functors Ll and 1:. K of normed vector spaces. 5. 5. The Aronszajn-Gagliardo Theorem Let A be an interpolation space with respect to A. It is natural to ask if there is an interpolation functor F, such that F(A) = A. This question is considered in the following theorem. 1. Theorem (The Aronszajn-Gagliardo theorem). Consider the category fJI of all Banach spaces. Let A be an interpolation space with respect to the couple A.