Show that the range of a bounded linear operator is not necessarily closed.
Hint: Use the linear bounded operator defined by .
Attempt to solution
My idea was to find an element that does not belong to the range and then try to build a convergent sequence in that has limit . The element satisfy the criteria because , with , but, clearly, , therefore, . The problem arise when I try to build the sequence, because with cannot converge to . Briefly, my problem is that I canīt find a limit point of that doesnīt belong to .