Isomorphisms of Banach Spaces

[jinydu]

If X and Y are algebraic categories, an isomorphism f:X->Y is a set bijection that preserves structure, and whose inverse set map also preserves structure. In general, the phrase "whose inverse..." is necessary. For example, there is an example of a bijection between topological spaces that is continuous, but whose inverse is not continuous.

However, there are special cases in which we can throw out the condition on the inverse. For instance, when proving that a map between two vector spaces is an isomorphism, it is enough to show that it is bijective and linear; linearity of the inverse follows automatically.

My question is whether this works for Banach spaces. I tried both ways (trying to prove the answer is yes, and trying to prove it is no); but couldn't finish either.

If f:X->Y is bounded linear and bijective but f^-1 is unbounded, then I showed that there is a sequence {x_n} in X with ||x_n||=1 such that f(x_n) converges to 0. If the unit sphere in X is compact, this gives a contradiction. But of course that isn't guaranteed in general...

I also tried constructing a counterexample. Let X = {{a_n}:sum_n |a_n|<infinity} and Y={{a_n}:sum_n |n a_n|<infinity}. Let f(a_n)=a_n/n. It is easy to see that f is bijective and bounded, while f^-1 is unbounded. Unfortunately, I'm not sure whether Y is complete (it is a normed vector space with norm ||{a_n}||=sum_n |a_n/n|).

Thanks