How wrong intuitions about weak* topology and completions of a normed space pose serious problems

Abstract

We learn mathematics subjectively and must apply it objectively. But sometimes, we apply it subjectively by using wrong intuitions which may be elusive to our eyes. The aim of this note is to disclose the secretes of two kinds of these false intuitions and the opportunities they may provide. We first discuss the wrong assumption which says that each topology is uniquely determined by studying a very bad phenomenon happening in dealing with weak* topology. Then, we consider the problem of completing a normed space under two comparable norms, one being smaller than the other. Here, we show that the common belief contending that smaller norms give rise to larger completions is wrong. We then pose some serious questions arising from these wrong intuitions. As we will see finding fallacies are as important as major mathematical activities like proving and disproving.