New Stone-Weierstrass Theorem

Without the successful work of Professor Kakutani on representing a unit vector space as a dense vector sub-lattice of ( ) ( ) , ⋅ C X in 1941, where X is a compact Hausdorff space and C(X) is the space of real continuous functions on X. Professor M. H. Stone would not begin to work on “The generalized Weierstrass approximation theorem” and published the paper in 1948. Latter, we call this theorem as “Stone-Weierstrass theorem” which provided the sufficient and necessary conditions for a vector sub-lattice V to be dense in ( ) ( ) , ⋅ C X . From the theorem, it is not clear and easy to see whether 1) “the vector sub-lattice V of C(X) contains constant functions” is or is not a necessary condition; 2) Is there any clear example of a vector sub-lattice V which is dense in ( ) ( ) , ⋅ C X , but V does not contain constant functions. This implies that we do need some different version of “Stone-Weierstrass theorem” so that we will be able to understand the “Stone-Weierstrass theorem” clearly and apply it to more places where they need this wonderful theorem.


Introduction
Throughout this paper, [T] <ω denotes the collection of all finite subsets of the given set T, "nhood" represents the word "neighborhood", C(Z) (or C(X)) is the space of real (or bounded real) continuous functions on compact Hausdorff space Z (or X), and ||⋅|| is the supremum norm; i.e., ( ) ( ) . For the other terminologies in Functional Analysis or General Topology which are not explicitly defined in this paper, the readers will be referred to the References [1] [2].Works on the sufficient and necessary conditions for a vector sub-lattice or vector sub-algebra V to be dense in were initiated in 1941 when Professor Kakutani tried to represent an order unit space V as a dense vector sub-lattice of ( ) ( ) , ⋅ C Z .It seemed that at that time Professor Kakutani did not know the sufficient and necessary conditions for a vector sub-lattice V to be dense in ( ) ( ) Professor Kakutani knew that a vector sub-lattice V is dense in V separates points of Z; and 2) V contains constant functions.In 1948, when Professor M. H. Stone published the "Generalized Weierstrass approximation theorem", as I know, he did give honor and credit to Professor Kakutani for the work in inspiring the paper M. H. Stone published in 1948.In my personal opinion, a) V separates points of Z and b) V contains constant functions are sufficient conditions for a vector sub-lattice V to be dense in ( ) ( ) . It seemed that it first appeared in Professor Kakutani's paper in1941.So, we should call this theorem as "Kakutani's theorem".Therefore, we will cite the Theorem 3.4 as Kakutani's Theorem in Section 3 and prove it with the results either in Section 2 or in a Theorem of Section 4.

A Characterization of Compact Sets
Due to the lack of original document in proving X in Section 3 is compact by Professor Kakutani.We insert this section as Section 2 to develop some necessary results for proving that X is a compact Haudorff space.Let A be a family of continuous functions Proof.Let { } j y be an A-net in F. For each y j , pick a net { } j i x in E converging to y j .
For each f α in A, by setting that ( ) ( )

( )
Cl f E is compact for each f in A, and 2) every A-net in E converges in Cl(E).

Kakutani Theorem
Definition 3.1 An element e in a vector lattice V is called an order unit if for every v in V, there is a r > 0 such that |v| ≤ re.Definition 3.2 A topological vector lattice V is called an order unit space if V contains an order unit e such that the topology on V is equivalent to the topology induced by the unit norm

{ }
Inf 0 Let L be the collection of all real continuous lattice homomorphisms t on the order unit space ( ) . Equip L with the weak topology induced by V. Then V is a space of real continuous functions on L. From now on, we regard every v in V as a real continuous function on L defined by ( ) ( ) for all t in L. It is obvious that: 1) V separates points of L: Since for any two different points s and t in L, s and t are two different real continuous lattice homomorphisms on ( ) , thus there is a v in V such that ( ) ( ) This implies that V separates points of L.
2) L is a Hausdorff space : Since the topology on L is the weak topology induced by V, V is a set of real continuous functions on L and V separates point of L, therefore, L is a , then X is a Hausdorff space.I believe that Professor Kakutani had proved that X is compact.We have no document available to see his proof.Let's prove it by Corollary 2.3 in this paper as the following: Theorem 3.3 X is a compact Hausdorff space.

( )
Cl X X = .Let's prove that X satisfies (1) and (2) in Corollary 2.3: 1) For each v in V, there is a n in ℕ such that v ne < , thus ( ) ( ) , it can be readily proved that r is a real lattice homomorphism on V such that ( ) 1 r e = ; i.e. r is in X and { }

Kakutani's Theorem
Theorem 3.4 Let V be a vector sub-lattice of C(X) such that 1) V separates points of X, and 2) V contains constant functions, then V is dense in Proof.We are going to show that for any f in C(X), any x, y in X and any ε > 0, there is a g in V such that ( ) ( ) .Then be a family of continuous functions f α on Y into Hausdorff spaces Y α such that the topology on Y is the weak topology induced by A. E, F two subspaces of Y such that Cl(E) is the closure of E in Y. Then the following are equivalent: 1) Every A-net in E has a cluster point in F. 2) Every A-net in F converges in F.
in Λ.Thus { } j y converges to x in F. (2) implying (1) is obvious. Theorem 2.2 Let A be a family of continuous functions on a topological space Y. Then Y is compact iff 1) f(Y) is contained in a compact subset C f for each f in A, and 2) every A-net has a cluster point in Y. Proof.Let { } i x be an ultranet in Y.For each f in A, an ultranet in C f , hence converges in C f ; i.e., { } i x is an A-net.(2) implies that { } i x has a cluster point x in Y. Since { } i x is an ultranet, { } i x converges to x.Thus, Y is compact.The converse is obvious. Corollary 2.3 Let A be a family of continuous functions on Y into Hausdorff spaces such that the topology on Y is the weak topology induced by A. E a subspaces of Y then Cl(E) is compact iff 1) ( ) will use the result of Stone-Weierstrass theorem (Theorem 4.1) to prove that V is dense in f x g x ε − < and ( ) ( ) f y g y ε − < : For any x and y in X, since V separates points of X, pick a k in V such that ( ) ( ) k x k y ≠