In this paper the nonstandard theory of topological vector spaces is developed, with three main objectives: (1) creation of the basic nonstandard concepts and tools; (2) use of these tools to give nonstandard treatments of some major standard theorems; (3) construction of the nonstandard hull of an arbitrary topological vector space, and the beginning of the study of the class of spaces which results. Introduction. Let 9 be a set theoretical structure and let *VR be an enlargement of Xll. Let (E, 0) be a topological vector space in VR. ??1 and 2 of this paper are devoted to the elementary nonstandard theory of (E, 0). In particular, in ?1 the concept of 0-finiteness for elements of *E is introduced and the nonstandard hull of (E, 0) (relative to *)1R) is defined. ?2 introduces the concept of 0-boundedness for elements of *E. In ?5 the elementary nonstandard theory of locally convex spaces is developed by investigating the mapping in *)R1 which corresponds to a given pairing. In ??6 and 7 we make use of this theory by providing nonstandard treatments of two aspects of the existing standard theory. In ?6, Luxemburg's characterization of the pre-nearstandard elements of *E for a normed space (E, p) is extended to Hausdorff locally convex spaces (E, 0). This characterization is used to prove the theorem of Grothendieck which gives a criterion for the completeness of a Hausdorff locally convex space. In ?7 a nonstandard proof is given of the theorem of Krein which states that, in a complete Hausdorff locally convex space, the weakly closed convex hull of a weakly compact set is again weakly compact. The nonstandard proof is somewhat simpler than the standard ones. In particular, we give a quite simple proof of the preliminary result (Corollary 7.3) which states ihat if A is a 0-compact set in a topological vector space (E, 0) and if {fnl is a sequence of 0-continuous linear functionals on E which is uniformly bounded on A and which converges to 0 on A, pointwise, then the sequence In} converges to 0, pointwise, on the closed convex hull of A. ?8 concerns the natural pairing between the nonstandard hull (E, p) of a normed space (E, p) and the nonstandard hull (E' p') of the dual space (E', p'). We give a standard condition on (E, p) (independent of *V) which is equivalent to each of Presented in part to the Society, March 31, 1972 under the title The dual space of the non standard hull of a normed space; received by the editors October 18, 1971. AMS (MOS) subject classifications (1970). Primary 02H25, 46Axx; Secondary 26A98, 54J05. Copyright ? 1973, American Mathematical Society 405 This content downloaded from 157.55.39.191 on Tue, 11 Oct 2016 05:21:01 UTC All use subject to http://about.jstor.org/terms 406 C. W. HENSON AND L. C. MOORE, JR. [October the following statements, provided that *31I is 1R-saturated: (i) the dual space of (E, p?) is (El, p'); (ii) (E, p) is reflexive. It is a consequence of this result that there is a reflexive Banach space whose nonstandard hull is a nonreflexive Banach space whenever *)R is ' l-saturated. On the other hand, we show that, under the same saturation assumptions, if 1 0. Let X be any set in X1I. We define *[XI to be the set of standard elements of *X; that is, *[X] = {*x I x C XI. If p is an element of *X, then Filx (p) is the ultrafilter on X determined by p: This content downloaded from 157.55.39.191 on Tue, 11 Oct 2016 05:21:01 UTC All use subject to http://about.jstor.org/terms 19721 NONSTANDARD THEORY OF TOPOLOGICAL VECTOR SPACES 407 Filx (p) = IYI Y C X and.p e *Y. (Filx (p) is written simply as Fil (p) when convenient.) If 5f is any collection of subsets of X which has the finite intersection property, then the filter monad of , ,[(0f), is defined by it(&) = nf*Al A f l[= n [X]. Recall that if *IJ1 is an enlargement of )11, then for each X in V11 and each collection 5f of subsets of X which has the finite intersection property, [0) 0. In fact, in that case there is an element A of *5f which satisfies A C [0). (Theorem 2.1.5(a) of [61.) Although we usually assume only that R1I is an enlargement of M1I, it is occasionally necessary, in order to achieve a smooth theory, to assume that *)J1 is also K-saturated (in the sense of [61) for a sufficiently large cardinal number K. This assumption will always be stated explicitly where needed. (When discussing the property of K-saturation, we will always assume that K is uncountable.) Recall that *)RJ is K-saturated if and only if for every X in M1I and every collection ( of internal subsets of *X, if ( has the finite intersection property and has cardinality less than K, then C3 has nonempty intersection. (Theorem 2.7.12 of [61.) Another useful property of R11, related to K-saturation, we choose to call Kenlarging (K an infinite cardinal). We say that *31I is a K-enlarging extension of 31I if and only if for every X in M1I and every collection ( of internal subsets of *X, if a has the finite intersection property and the number of elements of a which are not standard is less than K, then a has nonempty intersection. (Recall that a subset of *X is standard if it equals *Y for some subset Y of X.) We will be primarily interested in the property of being an No-enlarging extension of X1I. This property of *IR is equivalent to asserting that the three (equivalent) conditions in Theorem 2.7.3 of [61 hold in *31I Note that if K is a cardinal, then the direct limit of a chain (of order type K) of successive enlargements, beginning with V11, is a K-enlarging extension of X11. Also, if K is larger than the cardinality of every set in M1I, then *911 is K-saturated if and only if JIR is K-enlarging. Therefore, each structure M1I has K-saturated extensions and K-enlarging extensions for every cardinal number K. Throughout this paper E and F will denote vector spaces over R or C, usually assumed to be members of X11. For convenience we let K stand for either R or C. Thus *Ko is the set of finite elements of *K and *K1 is the set of infinitesimal elements of *K. If E is a K-vector space, then the addition on *E is denoted by + (as it is in E) and the scalar multiplication operation on *K x *E takes (X, p) to Xp. Now let E be a K-vector space. The algebraic dual of E (consisting of all linear functionals from E to K) is denoted by E4. If 0 is a vector topology on E, This content downloaded from 157.55.39.191 on Tue, 11 Oct 2016 05:21:01 UTC All use subject to http://about.jstor.org/terms 408 C. W. HENSON AND L. C. MOORE, JR. [October then the dual space of (E, 0) (consisting of all 0-continuous members of EN) is denoted by (E, 0)', or simply by E'. If E and F are K-vector spaces, then a pairing of E and F is a bilinear functional ( ...... ) on E x F which satisfies: (i) if x e E and x 0 0, then (x, y) 7 0 for some y C F, and (ii) if y C F and y 0 0, then (x, y) X 0 for some x C E. We denote the weak topology on E defined by F and the given pairing by a(E, F). Also, r(E, F) is the Mackey topology and /8(E, F) is the strong topology determined by the pairing. Let E be a K-vector space which is in V11 and let 0 be a vector topology on E. Denote by 11(0) the filter of 0-neighborhoods of 0, and let tO(O) be the unique translation invariant uniformity on E generating the topology 0. Then ((0) is the filter on E x E generated by the filter base of all sets of the form {(x,y)| x, y E E and x y E U} where U ranges over 11(0). Recall that for each x in E, p0(x) is defined by