令X、Y為緊緻的Hausdorff Spaces，且E、F為Banach Spaces。一線性映射T：C (X，E)→C (Y，F)為斥性如果∥Tf(y)∥∥Tg(y)∥＝0成立在對每個x屬於X，每個y屬於Y，∥f(x)∥∥g(x)∥＝0的條件下。高華隆、蔣志祥和黃毅青教授等證明了當h是一個由Y的函數到集合由E到F所構成可逆的線性算子且Tf＝hf○φ是由Y到X的同胚時，則雙保斥性映射T為一加權的組成算子：Tf＝hf○φ。在這篇論文中，我們延伸先前的結果至當Hausdorff Spaces為σ-緊緻或first countable時，連續函數可被定義到局部緊緻的Hausdorff Spaces。此外，我們對Mrcun最近做出來的結果給了一個簡短證明。最後，對於Araujo於二零零三年發表在Adv. Math期刊中關於光滑函數之雙保斥性映射的結果，我們試圖另闢蹊徑而達到相同結果。

Let X. Y be compact Hausdorff spaces, and E, F be Banach spaces. A linear map T：C (X，E)→C (Y，F) is separating if ∥Tf(y)∥∥Tg(y)∥＝0 whenever ∥f(x)∥∥g(x)∥＝0, for every x belonging to X, y belonging to Y. Gau, Jeang and Wong prove that a biseparating linear bijection T is a weighted composition oprator Tf＝hf○φ where h is a function from Y into the set of inveritable linear operators from E onto F and φ is a homeomorphism from Y onto X. In this thesis, we extend this result to the case that continuous functions are defined to a locally compact Hausdorff space, which is either σ-compact or first countable. Moreover, we give a short proof of a recent result of Mrcun. Finally, we give an alternative approach to an Araujo's result concerning biseparating maps of smooth functions appeared in Adv. Math.

Abstract
1. Introduction
2. Preliminaries and notations
3. Biseparating operators of continuous functions on a locally compact Haudsdorff space
4. Biseparating maps between spaces of vector-valued smooth functions
References

J. Araujo, Linear biseparating maps between spaces of vector-valued differectiable functions and automatic continuity, Adv. Math 187(2004), 488-520.

H. L. Gau, J. S. Jeang, and N. C. Wong, Biseparating linear maps between continuous vector-valued function spaces, J. Aust. Math. Soc. 74(2003), 101-109.