| dc.description.abstract | The notion of rotundity of norms is a very interesting area of research
in analysis. The question whether if a norm is Highly Rotund implies
it is Midpoint Locally Uniformly Rotund or whether a Uniformly Ro-
tund norm in Weakly Compact sets of direction implies Weakly Uniformly
Rotund is not clear. It remains an open question whether or not Mid-
point Locally Uniformly Rotund implies Highly Rotund. In this study, we
considered the Rotundity properties in a Fr´ echet space. The Rotundity
properties of major concern were Locally Uniformly Rotund, Midpoint
Locally Uniformly Rotund and Highly Rotund. The objectives of this
study therefore were to: Characterize Midpoint Locally Uniformly Ro-
tund norms; Determine the relationship between a Rotund norm and
Locally Uniformly Rotund norms; Establish the relationship between a
Rotund norm and a Highly Rotund norm. The methodology involved the
use of triangle inequality, method of modulus of convexity in estimating
the suprema and infima within the space and the method of renorming
Fr´ echet spaces to satisfy the rotundity conditions. The results show that
uniform and most rotundity exist in Fr´ echet spaces. It is also confirmed
that any equivalent norm on a Fr´ echet space which is Very Rotund is also
Rotund in Fr´ echet space. Moreover, it is shown that if a norm is Locally
Uniformly Rotund in a Fr´ echet space then it implies that it is Rotund
in a Fr´ echet space. The result obtained will be helpful in comprehensive
study on electromagnetic spectrum by quantum physicists and also in
determination of the periodic points of the orbit of chaotic operators on
Banach spaces. | en_US |