In Topics in Banach space theory by Albiac and Kalton, Definition 1.3.8 goes as follows:

Let $X$ and $Y$ be Banach spaces. We say two sequences $(x_n)_{n=1}^\infty\subseteq X$ and $(y_n)_{n=1}^\infty\subseteq Y$ are congruent with respect to $(X,Y)$ if there is an invertible operator $T:X\to Y$ such that $Tx_n=y_n$ for all $n\in\mathbb{N}$

Am I supposed to interpret that $T$ is bounded (and therefore also $T^{-1}$)? There are invertible non bounded operators, but later in the text it says: $T$ preserves every isomorphic property of $(x_n)_{n=1}^\infty$ and later something about $\Vert T\Vert\Vert T^{-1}\Vert$ and that can only be calculated if $T$ is bounded.



1 Answer 1


Some people use the term 'operator' for continuous linear maps. In this case I think $T$ is assumed to be continuous and bijective (so it has bounded inverse).


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