By Jean Dieudonné

A vintage to be had back! This e-book lines the heritage of algebraic topology starting with its construction by means of Henry Poincaré in 1900, and describing intimately the $64000 principles brought within the concept ahead of 1960. In its first thirty years the sphere appeared restricted to purposes in algebraic geometry, yet this replaced dramatically within the Thirties with the production of differential topology through Georges De Rham and Elie Cartan and of homotopy concept through Witold Hurewicz and Heinz Hopf. The impact of topology started to unfold to an increasing number of branches because it steadily took on a crucial position in arithmetic. Written via a world-renowned mathematician, this booklet will make fascinating studying for someone operating with topology.

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**Example text**

3. 6. Since these have the same underlying vector space, they clearly form a compatible couple. s. structures. s. on the space E E9p F (1 < p < (0) as the one obtained by interpolation, that is, we set E E9 p F = (Eo, Ede with () = lip. A similar argument applies for t'p ({Ei liE I}). The resulting operator space has many of the nice properties one would expect, for instance, we have a completely isometric identity (Eo E9p Ed* = E~ E9p ' Ei In interpolation theory, the intersection Eo n El and the sum Eo + El play an important role.

Structure, just like a normed space is simply a vector space equipped with a norm. However, it is not true that any sequence of norms on the spaces 1Ifn(V) can occur. s. structure on V. 1. Minimal tensor product Consider two operator spaces E c B(H) and Fe B(K). Then their minimal (or spatial) tensor product is defined as the completion of the algebraic tensor product E Q9 F with respect to the norm induced by B(H Q92 K) via the embedding E Q9 Fe B(H Q92 K). We denote by E Q9min F the resulting space and by II Ilmin its norm.

3. Let u: E -> F be a mapping between operator spaces. Show that for any aI, ... ,an in E we have II L u(aj)*u(aj)lIl/2 :s Ilullcbli L ajajll1/2 and II L u(aj)u(aj)*lll/2 :s Ilullcbli L ajajII1/2. 4. Let u: E -> F be a mapping between operator spaces. Show that Ilulicb = sup{llvullcb I v: F -> Mn Ilvllcb:S I}. 5. (Schur Multipliers) (i) Let {Xi I i :S n} and {Yj I j :S n} be elements in the unit ball of a Hilbert space I<. Then the mapping u: lIIn -> Mn defined by u([aij]) = [aij(xi, Yj)] is a complete contraction.