Inverse of operators norm Could someone help me to prove this inequality and decide if it can be an equality? Let be T a lineal isomorphism between $ (X,||·||) (Y,||·||)$ prove that $||T^{-1}||\geq||T||^{-1}$. Thanks.

We have $1=\|\operatorname{Id}\|=\|T^{-1}\circ T\|\leqslant\|T^{-1}\|.\|T\|$ and therefore $\|T^{-1}\|\geqslant\|T\|^{-1}$. However, in general the equality doesn't hold. Take, for instance, $X=Y=\mathbb{R}^2$ with its usual norm. Define $T\colon\mathbb{R}^2\longrightarrow\mathbb{R}^2$ defined by $T(x,y)=(2x,y)$. Then $\|T\|=2$ and $\|T^{-1}\|=1$.