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Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j \in\Bbb N\} =+\infty$. Then there is an $x$ (indeed a residual set of $x$'s) such that $\sup\{\lVert T_{jk} x\rVert : j \in\Bbb N\} =+ \infty$ for all $k$.

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Hint: argue by contradiction, and apply Baire categories theorem to $$F_{k,n}:=\bigcap_{j\in\Bbb N}\{x\in X,\lVert T_{j,k}(x)\rVert_Y\leqslant n\}.$$

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