The inverse of bounded operator?

Solution 1:

If You consider an invertible, i.e. bijective and bounded linear operator $A:X\rightarrow Y$, between two Banach-spaces (it´s important they are complete), then as a consequence of Baires category theorem A is open (open mapping theorem) and so $A^{-1}:Y \rightarrow X$ is continuous, i.e. bounded.

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