I was reading an article from AMM, titled,

A continuous bijection from $l_2 $ onto a subset of $l_2$ whose inverse is everywhere discontinuous.

In this he constructed the function $T:l_2\rightarrow l_1$ as $T(x)=$$(\sigma(x_1)x_1^2, \cdots,\sigma(x_i)x_i^2, \cdots )$

And shown this function to be bijective, continuous whose inverse is everywhere discontinuous.

My question is, What's the motivation behind this example? Why did the author find this example? What's the importance of this example?


Topological point of view. Any continuous bijection between compact Huasdorff spaces is a homeomorphism, that is its inverse is continuous. This example shows that compactness condition can not be dropped. Author's example shows necessity of compactness in a strong sense, because the inverse of $T$ is not merely discontinuous, its everywhere discontinuous.

Functional analytic point of view. By bounded inverse theorem any continuous linear bijection between Banach spaces is a linear homeomorphism. Again, the author's example shows necessity of linearity in that theorem, otherwise you can get inverse map which is discontinuous (and even more - discontinuous at every point).


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