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

.