Let X and Y be normed spaces and X compact. If T : X--->Y is a bijective linear operator. Show that $\displaystyle {T}^{-1} $ is bounded.
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Since, as the user above me has pointed out, you need to show effort I will tell you a more general fact that you may have seen before from topology (something you probalby have taken) which answers this question (with a little work) nicely "Let $\displaystyle f:X\to Y$ be a continous bijection where $\displaystyle X$ is compact and $\displaystyle Y$ Hausdorff. Then, $\displaystyle f$ is a homeomorphism."
Let $\displaystyle f:X\to Y$ be a continous bijection where $\displaystyle X$ is compact and $\displaystyle Y$ Hausdorff. Then, $\displaystyle f$ is a homeomorphism since $\displaystyle f$ is is homeomorphism then X and Y and homeomorphic.F^-1 is continuous which conculdes that F^-1 is bounded. Is that all i have to say??