_{1}

Without the successful work of Professor Kakutani on representing a unit vector space as a dense vector sub-lattice of
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
. From the theorem, it is not clear and easy to see whether 1) “the vector sub-lattice

*of*

**V****C(X)**contains constant functions” is or is not a necessary condition; 2) Is there any clear example of a vector sub-lattice

*which is dense in , 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.**

*V*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.,

Works on the sufficient and necessary conditions for a vector sub-lattice or vector sub-algebra V to be dense in

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 on a topological space Y. A net

Proposition 2.1 Let _{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

Proof. Let _{j}, pick a net _{j}. For each f_{α} in A, _{α} in Y_{α} and_{α}, there is an _{α}, α is in H}, there is an _{α}, _{2}. Then _{α} is Hausdorff and

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 _{f}, hence converges in C_{f}; i.e.,

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)

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

Let L be the collection of all real continuous lattice homomorphisms t on the order unit space

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

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 Hausdorff space. Let

Theorem 3.3 X is a compact Hausdorff space.

Proof. By the setting,

Next, we will use the result of Stone-Weierstrass theorem (Theorem 4.1) to prove that V is dense in

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

Notes:

1) A lot of textbooks of Functional Analysis listed The above theorem as the “Stone- Weierstrass Theorem”. I strongly disagree on it.

2) In my opinion, the above Theorem 3.4 should be named as Kakutan’s theorem. Because Professor Kakutani used the result of this theorem to represent an order unit space

Due to that the closure of a sub-algebra is a vector sub-lattice of C(X) (by Lemma 44.3 in the Reference [

Theorem 4.1. Stone-Weierstrass Theorem

Let Z be a compact Hausdorff space. A vector sub-lattice or a sub-algebra V of C(Z) is dense in

Theorem 4.2. New Version of Stone-Weierstrass

Theorem

Let Z be a compact Hausdorff space. A vector sub-lattice or sub-algebra V of C(Z) is dense in

To show the equivalence between Theorem 4.1 and Theorem 4.2, it is enough to show the equivalence between the following statements (A) and (B):

(A) for any f in C(Z), any x, y in Z and any ε with 0 < ε < 1, there is a g in V such that

(B) for any x in Z and any ε with 0 < ε < 1, there is a g in V such that

Proof. (A) Þ (B): Let h_{1} be the function in C(Z) such that _{x} in V such that

(B) Þ (A): Let

Theorem 4.3. Theorem 4.1 and Theorem 4.2 are equivalent.

Remark 4.4:

1) If the vector sub-lattice or sub-algebra V in the Theorem 3.4 contains constant functions, (without using Theorem 4.1) then let g be the function such that

2) It is also clear to get an example of a vector sub-lattice V that is dense in

Example. For each

1) The Stone-Weierstrass Theorem is a great and wonderful theorem. We provided new version of Stone-Weierstrass Theorem, simply trying to understand the theorem better and trying to obtain more applications to where it should be.

2) It must be a tough work for getting sufficient and necessary conditions for a vector sub-lattice V to be dense in

Wu, H.J. (2016) New Stone-Weierstrass Theorem. Advances in Pure Mathematics, 6, 943-947. http://dx.doi.org/10.4236/apm.2016.613071