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An example is given of a nonreflexive Banach space\(\tilde X\) that is uniformly nonoctahedral (or uniformly non-l1(3)), in the sense that there is a λ>1 such that there is no isomorphismT ofl1(3) into\(\tilde X\) for which $$\lambda ^{ - 1} \left\| x \right\|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \left\| {T(x)} \right\|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \lambda \left\| x \right\|{\text{ }}if x \in l_1^{(3)}$$ .

It is shown that a Banach space is super-reflexive if and only if the girth of its unit ball is greater than 4. Consequently, “girth greater than 4” is a property preserved under isomorphisms and duality.

The natural definition of orthogonality of elements of an abstract Euclidean space is that x ly if and only if the inner product (x, y) is zero. Two definitions have been given [11](2) which are equivalent to this and can be generalized to normed linear spaces, preserving the property that every twodimensional linear subset contain nonzero orthogonal elements. The definition which will be used here (Definition 1.2) has the added advantage of being closely related to the theories of linear functionals and hyperplanes. The theory and applications of this orthogonality have been organized in the following sections, which are briefly outlined: 1. Fundamental definitions. An element x is orthogonal to an element y if and only if j|x+kyjj > ||x|l for all k. This orthogonality is homogeneous, but is neither symmetric nor additive. 2. Existence of orthogonal elements. An element x of a normed linear space is orthogonal to at least one hyperplane through the origin, while for elements x and y there is at least one number a for which ax+y Ix or ||ax+yj| is minimum (Theorems 2.2-2.3). 3. Orthogonality in general normed linear spaces. The limits N?(x; y) =+lim,+.?|lnx+yil jjnxjj =liml0?o [||x+hyjj -lixil ]/hexistandsatisfyweakened linearity conditions. Also, x Iax+y if and only if N_(x; y) < -a||x

Several characterizations of weak-compactness are given for subsets of complete locally convex linear topological spaces and of Banach spaces. Some are new and some are generalizations of known facts.

A relatively easy proof is given for the known theorem that a Banach space is reflexive if and only if each continuous linear functional attains its sup on the unit ball. This proof simplifies considerably for separable spaces and can be extended to give a proof that a boundedw-closed subsetX of a complete locally convex linear topological space isw-compact if and only if each continuous linear functional attains its sup onX.

It has been conjectured that a closed convex subset C of a Banach space B is weakly compact if and only if each continuous linear functional on B attains a maximum on C [5]. This reduces easily to the case in which C is bounded, and will be answered in the affirmative [Theorem 4] after some preliminary results are established. Following suggestions by Namioka and Peck, the result is then generalized, first to weakly closed subsets of Banach spaces and then to weakly closed subsets of complete locally convex linear spaces. The original motivation for this conjecture was the knowledge that it is true when C is the unit sphere of a separable Banach space [2], which was later extended to arbitrary Banach spaces [3, Theorem 5]. Additional support was given the conjecture when V. L. Klee [5] proved a seemingly related theorem-namely, that if C is a bounded closed non-weakly-compact convex subset of a Banach space, then there is a decreasing sequence {K1} of nonempty closed convex subsets of C such that, for each x E C and each m E [0, 1), the set x + m(C x) meets only finitely many of the sets {Ki}. Iff is a continuous linear functional whose sup on C is M and if there is no x in C for which f(x) = M, then a suitable choice for Klee's sequence {Ki} is to let Kn = C r1 {x:f (x) > M l/n} for each n. It is interesting to note that the conjecture can be verified easily for a bounded closed convex set that is symmetric about an interior point x, since if x is translated to 0, then the convex set as a unit sphere induces a norm for which the new Banach space is isomorphic to the original space. More generally, if 0 is an interior point of a bounded convex set C and K is the closed convex span of Cu ( C), then K is symmetric about 0 and the sup on K of a continuous linear functional is the larger of its sups on C and C. Therefore for convex bodies the conjecture can be established by using the known theorem for unit spheres [3, Theorem 5, p. 215]. The following theorem is a generalization of a characterization of weak compactness of the unit sphere that was useful in [2]. In the proof of this theorem and thereafter, we shall use the convention that a sequence of nonoverlapping

The purpose of this paper is to develop characterizations of weakly compact subsets of a Banach space in terms of separation properties. The sets A and B are said to be separated by a hyperplane H if A is contained in one of the two closed half-spaces determined by H, and B is contained in the other; A and B are strictly separated by H if A is contained in one of the two open half-spaces determined by H, and B is contained in the other. The following are known to be true for locally convex topological linear spaces.

The immediate postamputation plaster prosthesis was supplied as a central community service in a large metropolitan area in 75 lower-extremity amputations. The overall healing rate was 90%. Half the patients were ready for discharge in four weeks or less; 85% eventually walked with a prosthesis. Although this record is satisfactory, the spread of responsibility for the 75 amputations among 53 different surgeons forced the conclusion that, because of difficulties in communication, a central service of this type was not practical in that community as now constituted. Where continuity of the whole patient care team could be established, the value of the immediate postamputation plaster was confirmed. Experience with various types of going-home prostheses emphasized the need in this and other communities for facilities to enable safe provision of early, total-contact prostheses.

Tlhe first section of this paper will be devoted to the study of two particular surfaces. The methods used to treat thiese surfaces will then be extended to develop a classificationof surfaces, each class consisting of combinatorially eauivalent surfaces. Applications will be made to tlhe study of the topological nature of covering surfaces and of Riemann surfaces in particular. It will be assumed that the reader is familiar with the material of Chapter V, pages 235-244 and 256-264, in the book What Is 4lathsrmatics? by Courant and Robbins. Other than for references to this book, no previous knowledge of Topology will be assumed.