Abstract
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 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 , 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.
Highlights
Throughout this paper, [T]
It is clear that Professor Kakutani knew that a vector sub-lattice V is dense in (C ( Z ), ⋅ ) if 1) V separates points of Z; and 2) V contains constant functions
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
Summary
Throughout this paper, [T]
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