By Yitzhak Katznelson and Yonatan R. Katznelson
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Additional info for A (Terse) Introduction to Linear Algebra (Student Mathematical Library, Vol. 44)
E. q(min(1, t)). q(min(1, t)) = 1/(ql/q(8(1 - 8))1/q), we obtain (6). iA) c 1:(A). 1. The K-Method It remains to prove that KO,q is an exact interpolation functor of exponent Thus, suppose that T:A--+B, where A=(Ao,A I ) and B=(Bo,B I ). Put e. Then K(t, Ta; B)~infa=ao+al (II Taoll Bo +t II Tal liB) ~infa=ao+al (Mo Ilao IIAo + tM I IlalII A). Thus But, using (7) with s=Mo/MI' we obtain This proves that KO,q is an exact interpolation functor of exponent e. 0 Remark: The interpolation property holds for all operators T: l'(A)--+l'(B), such that (8) holds.
1. Theorem (The duality theorem). Let A be a couple of Banach spaces, such that A(A) is dense in Ao and Ai' Assume that 1 ~ q < 00 and 0 < () < 1. Then where 1/q + 1/q' = 1. 1. In order to prove (3), we take... a'e(Ao, A i )6,q;J' and apply Formulaj1). 8. l-O,q' are dual via the duality Lvrvocvf3v and e is arbitrary, (3) follows. ,v ~ a'v with convergence in 1:(A') = A(A)'. Then it follows that Since we conclude that which implies (4). 8. 1, we see that if AlcAO then A01,q1CAoM1 when (JO<(JI' and Ao,q c Ao,r when q ~ r.
4') and a'=a~+a~,a;EA;. 8. , II(a)1 ~ III II + = , aeLl(A). By the density assumption, a~ and a'i are determined by their values on Ll(A). Putting I = a~ + a~, 11111 1"(A') ~ 11111
A (Terse) Introduction to Linear Algebra (Student Mathematical Library, Vol. 44) by Yitzhak Katznelson and Yonatan R. Katznelson