Question
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
.