Linear mappings are fairly well-behaved
Classically, it is absurdly trivial that if an element x of a normed space E is not in the kernel of a linear mapping T, then Tx\(\neq 0\). Constructively, however, a stronger, and affirmative, meaning is given to the conclusion Tx\(\neq 0\), and it thereby becomes difficult to prove. In the constructive development of normed space the condition Tx\(\neq 0\) is taken to mean that \(\| Tx\| >0\), where the notion of positive real number is used in the strong constructive sense of explicit positive rational approximations. Typically, the hypotheses are also strengthened accordingly. The main result of the paper (expressed in the title) is that if T maps E onto a Banach space, and if \(\| x-y\| >0\) whenever \(y\in \ker (T)\), then Tx\(\neq 0\). Other results involve the graph of T, with applications to the boundedness of mappings onto finite-dimensional spaces.
scientific article; zbMATH DE number 4177919