By Nigel Ray, Grant Walker

J. Frank Adams had a profound impact on algebraic topology, and his paintings keeps to form its improvement. The overseas Symposium on Algebraic Topology held in Manchester in the course of July 1990 used to be devoted to his reminiscence, and nearly all the world's major specialists took half. This quantity paintings constitutes the lawsuits of the symposium; the articles contained the following variety from overviews to reviews of labor nonetheless in growth, in addition to a survey and entire bibliography of Adam's personal paintings. those lawsuits shape a tremendous compendium of present examine in algebraic topology, and one who demonstrates the intensity of Adams' many contributions to the topic. This moment quantity is orientated in the direction of homotopy concept, the Steenrod algebra and the Adams spectral series. within the first quantity the subject is especially volatile homotopy concept, homological and specific.

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**Example text**

E11 }. Let 4J be the family of all F-norms defined on :l(N) and let 4Jn be the family of all F-norms defined on R 11 • Now we shall determine the distance between two F-norms defined on an F-space X as follows. Letp(x), q(x) be two F-norms on X.

Let {xn} be a dense sequence in the space e 0(D). Let for t,t' E Q A A 00 d(t t') ' = ~ _1 lxn(t)-xn(t')! J 211 l+lxn(t)-xn(t')l' n=l It is easy to verify that d(t, t') is a metric. 7 we can show that the metric d(t, t') induces a topology equivalent to the original one. D Let (Xt, II lit) be a sequence of F*-spaces. Let X= (Xt)(s) be the space of all sequences x = {Xt Xt EXt}. The topology in X is given by a sequence of pseudonorms llxll; = llxtllt. Of course the space X is an F*-space. 9. It the spaces (Xt, II lit) are separable, then the space X= (Xt)(s) is separable.

Let {Yn} be a dense sequence in X. Let en = {0, ... , 0,1,0, ... }. Let {xn, m} be the sequence {en} enumerated n·th place as a double sequence. Let Zm,n = (yn,+Xn, m)· Let X 0 = lin{zn,m, n = 1, 2, ... , m = 1,2, ... }. }; tn,m n,m=! Zn,m. Therefore, if z o:j::. 0, then the natural projection of z on l 2 ,Pz, is different from zero. This implies, by the definition of the norm in the product of spaces, that inf suplltzll ~ z;eo teR zeXxN inf suplltPzll = +oo. z;e 0 teR zeXxN Thus X 0 is a {JF*-space.