My Ph. D. Thesis, written under the direction of Ky Fan at the University
of California at Santa Barbara, examined certain properties characterizing
the map that associates with a compact space X, the space C(X) of
continuous scalarvalued functions on X. At the same time, a continuous
function f from X to Y is mapped to the operator T from C(Y) to C(X)
defined by T(g) = g composed with f. My thesis was published in Studia
Mathematica a lifetime ago. An elementary exposition of the properties
of the functor described above can be found in An
Important Functor in Analysis and Topology, American Math Monthly,
85(1978). Subsequent
research was devoted to relationships between X and C(X), an area that
was extremely popular in the 1960s. In particular, connectedness properties
of X can be characterized in various ways. See Local
connectedness and pseudocompactness in completely regular spaces, Proc
Amer Math Soc, 68(1978) and A
Banach space property of C(X) characterizing local compactness in X,
Pure and Applied Math Sci, 12(1980). I also published an interesting
article entitled Cones in LSpaces in the Bulletin of the Instutute
of Mathematics of the Academia Sinica (Vol 2 No 2, 1974). This was
dedicated to my advisor in the occasion of his 65th birthday.
For work more closely related to my
teaching see The
Riesz Representation Theorem Revisited, American Math Monthly, 90(1983), On
the Differentiation Formula for Sine, American Math Monthly, 96(1989),
and L’Hopital’s
Rule via Integration, American Math Monthly, 98(1991).
In the late 1980s and early 1990s I became intensly interested in
measure theory, presenting an occasional paper and working on questions
associated with the validity of the Lebesgue Convergence Theorem in
a general topological measure space. At about the same time I became
an active participant in the Calculus reform movement investigating
the possibilities associated with advances in computer software and
hand calculators. See PROFESSIONAL ACTIVITIES.
