By M.A. Naimark

book and to the writer NOORDHOFF who made attainable the looks of the second one version and enabled the writer to introduce the above-mentioned modifi cations and additions. Moscow M. A. NAIMARK August 1963 FOREWORD TO the second one SOVIET version during this moment variation the preliminary textual content has been labored once more and better, convinced parts were thoroughly rewritten; specifically, bankruptcy VIII has been rewritten in a extra available shape. The alterations and extensions made through the writer within the eastern, German, first and moment (= first revised) American, and likewise within the Romanian (lithographed) variants, have been hereby taken under consideration. Appendices II and III, that are useful for realizing bankruptcy VIII, were integrated for the ease of the reader. The publication discusses many new theoretical effects that have been constructing in tensively through the decade after the book of the 1st version. in fact, lim itations at the quantity of the publication obliged the writer to make a difficult choice and in lots of circumstances to restrict himself to easily a formula of the hot effects or to mentioning the literature. the writer used to be additionally forced to select of the highly broad selection of new works in extending the literature record. Monographs and survey articles on certain themes of the speculation which were released up to now decade were integrated during this checklist and within the litera ture mentioned within the person chapters.

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**Sample text**

Consequently, Fp has a maximal element F which, in virtue of the preceding lemma, is defined on the entire space R. COROLLARY I. If P is a Minkowski functional in the real linear space R, then for arbitrary vector Xo E R there exists a linear functional F, defined on all of Rand satisfying the conditions p(x o), (8) p(x)for all x E R. (7) F(xo) F(x) ~ = Proof. We shall denote by mthe set of all vectors IXX 0, where runs through all real numbers; ID1 is a subspace of R. We define a linear functionalfin ID1 by means of the formula (9) so that IX (10) We shall prove that f satisfies condition (6); on the basis of Theorem 1 it can then be extended to a linear functIOnal F defined on all of R and satisfying condition (7); in virtue of (10) condition (8) will then be satisfied also, and Corollary 1 will be proved.

Proof. Suppose F I , F2 are closed (and hence compact) disjoint sets in X. By I, subsection 7. for any point y E F2 there exist disjoint open sets Uy , Vy , which contain F[ and y, respectively. Among the sets Vy there is a finite number Vy1 , Vy2 , .. " VVn which form a covering of F 2 . Then n n U = k= 1 n U Yk' V = UV k= 1 Yk are disjoint open sets, containing Fl and F 2 , respectively. n (URYSOHN'S LEMMA). For any two disjoint closed subsets F 0, Fl of a normal space X there exists a continuous real-valued function f on X satisfying the conditions: l)O~f(x):£I: 2) f(x) = 3) f(x) = 0 on Fo: I on F[ .

U(x~J through all possible neighborhoods of the points X~l' .. " x~n' we obtain a family of sets {U(XO)} which we shall consider to be a neighborhood basis of the point xo. l XIX becomes a topological space. It is called the topological product of the spaces XIX' If X = (XIX)' then XIX is called the IX-th coordinate of the point X and the projection of the point X on XIX' The correspondence x -+ XIX is a mapping of the space X = m §2] TOPOLOGICAL SPACES 35 rr~e~ X" onto X,,; the image of a set M c X under this mapping is called the pro- jection of M on X".