It is known that a Uniformly Rotund norm implies a Locally Uniformly Rotund norm. The question whether if a Fre’chet space F has a Rotund norm implies it has an equivalently Locally Uniformly Rotund norm is still open and represents one of the most interesting and studied problems. In this paper, we investigate if there exists a direct relationship between a Rotund norm and a Locally Uniformly Rotund norm in Fre’chet space. It is shown that if a norm is Locally Uniformly Rotund in a Fre’chet space then it implies that it is Rotund too in a Fre’chet space. It is also shown that if a Fre’chet space is non-reflexive such that its dual is separable then the norm defined on it is an equivalent norm which is Rotund hence Locally Uniformly Rotund. It is further shown that any separable Fre’chet space that admits an equivalent Locally Uniformly Rotund norm must admit a Rotund norm.

Keywords : A Norm; Rotund Norm; Locally Uniformly Rotund Norm; Fre’chet Space; Non-Reflexive Fre’chet Space.