Stone-Weierstrass Research Articles

Separation and Approximation in Topological Vector Lattices

Spectral theory in its lattice-theoretic setting proves abstractly that the indicators of measurable sets generate the space L of Lebesgue-integrable functions on an interval. We are concerned here with abstractions suggested by the fact that indicators of intervals suffice to generate L. Our results show that the approximation of arbitrary elements of a topological vector lattice rests upon the ability to separate disjoint elements/ and g by an operation that behaves in the limit like a projection annihilating/ and leaving g invariant.The introduction of this concept of separation together with the notion of limit unit leads (via the Fundamental Lemma) to abstract generalizations of the Radon-Nikodym Theorem (Theorem 1) and the Stone-Weierstrass Theorem (Theorem 3).

On the Katětov and Stone-Čech Extensions

In general topology, one knows several standard extension spaces defined for one class of spaces or another and it is a natural question concerning any two such extensions which are defined for the same space whether they can ever be equal to each other. In the following, this problem will be considered for the Stone-Čech compactification βE of a completely regular non-compact Hausdorff space E[4] and Katětov's maximal Hausdorff extension κE of E[5]. It will be shown that βEκE always holds or, what amounts to the same, that κE can never be compact. As an application of this it will be proved that any completely regular Hausdorff space is dense in some non-compact space in which the Stone-Weierstrass approximation theorem holds.

Algebras of Uniformly Continuous Functions

There is a well-known dual relationship between compact Hausdorff spaces X and the algebras C(X) of all real-valued continuous functions on the spaces. This generalizes, imperfectly of course, to completely regular spaces X; the algebras C(X) are no longer Banach algebras, and it is not too well understood what they are. This paper concerns a slightly more general class of algebras C(pX) corresponding dually to certain' uniform spaces pX. (One obtains C(X) by choosing ,p so that all continuous functions are uniformly continuous.) Most of the known results for unbounded function algebras C(X) go over, and in generalizing we obtain two main advantages. The first main advantage is an analogue of the Stone-Weierstrass theorem. That theorem tells us that the algebras C(X), for X compact, are just the algebras of bounded real-valued functions which are complete under uniform convergence. The present algebras C(pX) are those algebras of real-valued functions which are complete under uniform convergence and closed under all continuous n-ary operations; that is, if Al -,fn are in the algebra and g is a continuous real-valued function on En, then g(f1, *-,ffn) is in the algebra. (Also, any subalgebra closed under all operations and inducing the given uniformity is uniformly dense. No analogous results are known for general C(X).) The second main advantage is an analogue of Tietze's extension theorem: every uniformly continuous real valued function on a subspace of a uniform space, of the special class here considered, can be extended over the whole space. Then by considering only complete spaces one has a perfect bicategorical duality between the spaces fiX and the algebras C(pX), subspaces corresponding to quotient algebras, and quotient spaces to subalgebras. (Categorical duality means a suitable correspondence between algebra homomorphisms and continuous mappings, and the term bicategorical covers the suband quotient-systems. The latter concept was first formalized by MacLane [20], but a slightly different treatment is developed from scratch in this paper.) * The author is a National Science Foundation fellow. The work was advanced by, and reported on to, the 1955 National Science Foundation-American Mathematical Society Summer Institute on Set Theoretic Topology. Defined below. 96

On the Weierstrass approximation theorem

On the Weierstrass approximation theorem

A note on the Stone-Weierstrass theorem for quaternions

1. S. A. Amitsur, The identities of PI-rings, Proc. Amer. Math. Soc. vol. 4 (1953) pp. 27-34. 2. , Algebras over infinite fields, Proc. Amer. Math. Soc. vol. 7 (1956) pp. 35-48. 3. I. Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. vol. 54 (1948) pp. 575-580. 4. W. Krull, Jacobsonsches Radical und Hilbertscher Nullstellensats, International Congress of Mathematicians, Cambridge, 1950, Conference in Algebra, pp. 56-64.

Certain generalizations of the Weierstrass approximation theorem

Certain generalizations of the Weierstrass approximation theorem

Analytic approximations to topological transformations

A continuous transformation of a two-dimensional region into itself, or into a second such region may be characterized by a pair of continuous functions X= X(x,1y), Y= Y(x,1y). The continuity of these functions, and the existence of a unique pair of inverse functions are the only restrictions imposed by the topologist. On the other hand, the differential geometer usually demands of the transformations he considers, that the functions defining them be analytic. It is a matter of some interest to investigate the relations connecting these two types of transformation, and in particular to determine whether the general transformations of the first type may not be regarded as in some sense limiting cases of the more special analytic type. This question is here discussed, and we shall prove that every continuous one-to-one transformation of a two-dimensional region of finite connectivity may be approximated to an arbitrary degree of exactness by an analytic one-to-one transformationt The analogy between the present investigation and the Weierstrass approximation theorem is obvious. The peculiar difficulty of the problem here treated is the need of keeping our approximating transformation one-to-one. This requires a discussion of a combinatorial nature.

The Weierstrass Approximation Theorem

Journal of Mathematics and PhysicsVolume 4, Issue 1-4 p. 148-152 Article The Weierstrass Approximation Theorem† Philip Franklin, Philip FranklinSearch for more papers by this author Philip Franklin, Philip FranklinSearch for more papers by this author First published: January 1925 https://doi.org/10.1002/sapm192541148Citations: 1 †Presented to the American Mathematical Society, December 30, 1924. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume4, Issue1-4January 1925Pages 148-152 RelatedInformation