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Let $E$ and $F$ two normed vector spaces, $A \subset E$ compact, $B \subset F$ and $f: A \to B$ is a bijective continuous function. As $f$ is bijective, we can defining the inverse function $f^{-1} : B \to A$ by $$f^{-1}(y)=x \iff f(x)=y.$$ Show that $f^{-1}$ is continuous.

Theorem : $f$ is continuous $\iff$ for each open set $V$ in $B$, $f^{-1}(V)$ is open in $A$.

I think is preferable to use this theorem instead of the sequentially continuous theorem.

I am stuck on this problem. Is anyone is able to help me a bit to continue this question?

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This is just a special case of a general fact about topology: If $X$ is compact, $Y$ is Hausdorff, and $f:X\to Y$ is a continuous bijection then $f^{-1}$ is continuous:

Since $f$ is a bijection, the inverse image of $S$ under $f^{-1}$ is just $f(S)$. So you have to show that $f(S)$ is open for every open set $S\subset X$. Since $f$ is a bijection this is the same as showing that $f(C)$ is closed for every closed set $C\subset X$. But $X$ compact, $C$ closed implies $C$ is compact, so $f(C)$ is compact, and now $Y$ Hausdorff implies $f(C)$ is closed.

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Hint: show that $f(C)$ is closed for every $C$ closed in $A$.

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