By Jesus M.F. Castillo, Manuel González

This ebook on Banach house concept specializes in what were known as three-space difficulties. It includes a quite entire description of principles, tools, effects and counterexamples. it may be thought of self-contained, past a path in sensible research and a few familiarity with smooth Banach area equipment. it will likely be of curiosity to researchers for its tools and open difficulties, and to scholars for the exposition of recommendations and examples.

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The following method to obtain 3SP ideals starting from 3SP ideals was introduced in [269]. i. The functor co: X--, X c° = X**/X is exact. Proof. Let 0 -, Y--, X--, X / Y - , 0 be a short exact sequence. The proof is based in the isomorphisms (see [70, 269]): Y**/Y = (X+ Y ± ± )/X and (X/Y)**/(X/Y) = X * * / ( X + Y± ± ) which immediately give the exactness of the sequence Exact functors 51 0 --, Y * * I Y - , X * * / X - , (X/Y)**I(X/Y)--, O. [] Therefore. j. Given a 3SP ideal A, the residual ideal A c° is a 3SP ideal.

A twisted sum of Banach spaces Y ~ F Z is locally convex if and only if F is O-linear. Proof. Assume that the twisted sum Y ~ F Z is locally convex. It therefore coincides with the Sharked sum Y ~ C Z where C is the absolutely convex hull of all points (y, 0) with II y II = 1 and (F(z), z) with l1 z l[ < 1. Let z 1..... z n be elements of Z such that ]]l_<_i

1 + II z II is a norm if and only if [2 is pseudolinear. Proof. " Conversely, if II y - a z since, when X is a scalar, II + 11z II is a norm then [2 is homogeneous II X[2z - [2(Xz) II + II xz II = I1 (X[2z, Xz) II = I x l It (az, z) II = I x111 z II which yields Xflz = fl(kz). That [2 is pseudo-linear is very easy to verify: II [2z+aa-a(z+a)II + II z+b II = II (az+[2b, z+b)II --- II ([2z, z) II + II (ab, b) tl = Ilzll + Ilbll. [] Twisted sums and extensions 27 All this gives a complete characterization of the locally convex character of twisted sums in terms of quasi-linear maps [48].