Research - Dr. Paul C.-S. Lin
Dr. Paul C.-S. Lin
Department of Mathematics
Dr. Lin's current research interest is about finding new characteristic properties for Hilbert space from normed linear spaces, with some applications. This is a classical question and essential in the sense that most beautiful and interesting theorems in Functional Analysis are based on Hilbert spaces. Let X be a normed linear space over the real line. One can characterize an inner product space by means of orthogonalities in X. James (1945) defined and studied two types of orthogonalities. By using relations among orthogonal vectors he was able to prove that in X if either orthogonality has homogeneity or additivity property, then X is an inner product space. In this article we first define a new notion of (a,b,c,d)-orthogonality in an infinite dimensional normed linear space, which is natural extension of the two orthogonalities by James. It is proved that X is an inner product space if and only if one orthogonality implies the other. Our unified approach is different from James', Day's (1947), Freese (1983), or many others. Some applications of our results regarding inner product spaces and related topics in normed spaces are given.
Dr. Lin's published research articles involve the following topics of Pure Mathematics:
- Operator Theory (Bounded linear transformations on Hilbert spaces).
- C*-algebra.
- Operator Inequalities on Hilbert spaces.
- Inner Product and 2-Inner Product spaces with applications.
- Functional Analysis, Banach spaces in particular.
- Numerical Ranges and Spectral Theory.
