3. Prove that in the open mapping theorem (6.5.2) it is enough to assume, rather than T onto, that the range of T is of second category in Y , that is, it is not a countable union of nowhere dense sets.
4. Let (X, I·I) and (Y, |·|) be Banach spaces. Let Tn be a sequence of bounded linear operators from X into Y such that for all x in X, Tn (x ) converges to some T (x ). Show that T is a bounded linear operator.
5. Show that T in Problem 4 need not be bounded if the sequence Tn is replaced by a net {Tα }α∈I . In fact, show that every linear transformation from X into Y (continuous or not) is the limit of some net of bounded linear operators. Hint: Show, using Zorn's lemma, that X has a Hamel basis B, that is, each point in X is a unique finite linear combination of elements of B. Use Hamel bases to show that there are unbounded linear functions from any infinite-dimensional Banach space into R.